Classes of Harmonic Functions Defined by the Carlson–Shaffer Operator

Abstract

The Carlson–Shaffer operator plays an important role in the geometric theory of analytic functions. It is associated with the hypergeometric function and the incomplete beta function. The Carlson–Shaffer operator generalizes various other linear operators, such as the Ruscheweyh derivative operator, the Bernardi–Libera–Livingston operator, and the Srivastava–Owa operator. Ideas in the theory of analytic functions are often symmetrically transferred to the theory of harmonic functions. By using the Carlson–Shaffer operator, we introduce a class of harmonic functions defined by weak subordination. Next, we give some necessary and sufficient coefficient conditions for the class of functions. Furthermore, we determine coefficient estimates, distortion bounds, extreme points, and radii of starlikeness and convexity for the defined class.

1. Introduction

Linear operators play an important role in the theory of analytic functions. They are often symmetrically transferred to the theory of harmonic functions (see, for example, [1,2,3,4,5,6,7,8]). One of the most important linear operators is the Carlson–Shaffer operator, which is associated with the hypergeometric function. Let $\eta ,\theta ,\vartheta$ be complex numbers with $\vartheta \ne 0,-1,-2,\dots ,$ and let

$$(\eta ,k):=\left\{\begin{array}{ccc}1& \mathrm{for}& k=0\\ \eta (\eta +1)\dots (\eta +k-1)& \mathrm{for}& k\in \mathbb{N}\end{array}\right.$$

denote the Pochammer symbol. If $\eta \ne 0,-1,-2,\dots ,$ then

$$(\eta ,k)={\displaystyle \frac{\Gamma (k+\eta )}{\Gamma \left(\eta \right)}}$$

where $\Gamma$ denotes the gamma function. By using the Pochammer symbol, we define the hypergeometric function, ${}_{2}F_1(\eta ,\theta ;\vartheta ;\zeta )$ by

$${}_{2}F_1(\eta ,\theta ;\vartheta ;\zeta ):=\sum _{k=0}^{\infty }{\displaystyle \frac{(\eta ,k)(\theta ,k)}{(\vartheta ,k)}}{\displaystyle \frac{{\zeta }^k}{k!}},$$

(1)

It is well known that the power series (1) converges in the unit disk $\Delta :=\Delta \left(1\right),$ where $\Delta \left(\rho \right):=\{$ $\zeta$ $\in \mathbb{C}:\left|\zeta \right|<\rho \}$. For ℜe $\vartheta >\Re$e $\theta >0$, the hypergeometric function (1) has the following integral representation:

$${}_{2}F_1(\eta ,\theta ;\vartheta ;\zeta )={\displaystyle \frac{\Gamma \left(\vartheta \right)}{\Gamma \left(\theta \right)\Gamma (\vartheta -\theta )}}{\int }_0^1{t}^{\theta -1}{(1-t)}^{\vartheta -\theta -1}{(1-t\zeta )}^{-\eta }dt.$$

Now, we define the incomplete Beta function $\Phi (\eta ,\vartheta ;\zeta )$ as follows

$$\Phi (\eta ,\vartheta ;\zeta )=\sum _{k=0}^{\infty }{\displaystyle \frac{(\eta ,k)}{(\vartheta ,k)}}{\zeta }^{k+1},\zeta \in \Delta .$$

It is easy to see that

$$\Phi (\eta ,\vartheta ;\zeta )={\zeta }_2{F}_1(\eta ,1;\vartheta ;\zeta ).$$

If ℜe $\vartheta >\Re$e $\eta >0,$ then the function $\Phi (\eta ,\vartheta ;\zeta )$ may be written in the following form:

$$\Phi (\eta ,\vartheta ;\zeta )={\displaystyle \frac{\Gamma \left(\vartheta \right)}{\Gamma \left(\eta \right)\Gamma (\vartheta -\eta )}}{\int }_0^1{\displaystyle \frac{t^{\eta -1}{(1-t)}^{\vartheta -\eta -1}}{1-t\zeta }}dt.$$

Using the incomplete Beta function $\Phi (\eta ,\vartheta ;\zeta )$, Carlson and Shaffer [9] (see also [10]) defined the linear operator $\mathcal{L}(\eta ,\vartheta )$ on the class $\mathcal{A}$ of analytic functions via the following convolution

$$\mathcal{L}(\eta ,\vartheta )\varphi \left(\zeta \right)=\Phi (\eta ,\vartheta ;\zeta )\ast \varphi \left(\zeta \right),\varphi \in \mathcal{A}.$$

In particular, for $\vartheta >\eta >0$, we have

$$\mathcal{L}(\eta ,\vartheta )\varphi \left(\zeta \right)={\displaystyle \frac{\Gamma \left(\vartheta \right)}{\Gamma \left(\eta \right)\Gamma (\vartheta -\eta )}}{\int }_0^1{u}^{\vartheta -2}{(1-u)}^{\vartheta -\eta -1}\varphi \left(u\zeta \right)du.$$

It is clear that the Carlson–Shaffer operator maps a function $\varphi$ on the polynomial for $\eta \in \left\{0,-1,-2,\dots \right\}$. If $\eta \notin \left\{0,-1,-2,\dots \right\},$ then the Carlson–Shaffer operator maps the space $\mathcal{A}$ infectively onto itself. Moreover, $\mathcal{L}(\eta ,\eta )$ is the identity operator, and $\mathcal{L}(\eta ,\vartheta )$ is the inverse of $\mathcal{L}(\eta ,\vartheta ).$ Furthermore,

$$\mathcal{L}(2,1)\varphi \left(\zeta \right)=\zeta {\varphi }^{\prime }\left(\zeta \right),\mathcal{L}(1,2)\varphi \left(\zeta \right)={\int }_0^{\zeta }{\displaystyle \frac{\varphi \left(u\right)}{u}}du.$$

Since

$$\underset{n\to \infty }{lim}\sqrt[n]{\left|{\displaystyle \frac{(\eta ,k)}{(\vartheta ,k)}}\right|}=1,$$

the infinite series for $\mathcal{L}(\eta ,\vartheta )\varphi$ and $\varphi$ has the same radius of convergence. The Carlson–Shaffer operator generalizes other linear operators. Ruscheweyh [11] introduced the operator ${\mathcal{D}}^n:$ $\mathcal{A}\to \mathcal{A},$ defined by

$${\mathcal{D}}^n\varphi \left(\zeta \right)={\displaystyle \frac{\zeta {\left({\zeta }^{n-1}\varphi \left(\zeta \right)\right)}^{\left(n\right)}}{n!}}(n\in {\mathbf{N}}_0).$$

(2)

Next, we recall the generalized Bernardi–Libera–Livingston integral operator $J_{\nu }:\mathcal{A}\to \mathcal{A},$ defined by (cf. [12,13,14])

$$J_{\nu }\varphi \left(\zeta \right)={\displaystyle \frac{\nu +1}{{\zeta }^{\nu }}}\underset{0}{\overset{\zeta }{\int }}{t}^{\nu -1}\varphi \left(t\right)dt(\nu >-1;\varphi \in \mathcal{A}).$$

(3)

We observe that

$${\mathcal{D}}^n\varphi =\mathcal{L}(n+1,1)\varphi ,J_{\nu }\varphi \left(\zeta \right)=\mathcal{L}(1+\nu ,\nu +2).$$

(4)

Now, we recall here the fractional derivative operator $D_{\zeta }^{\lambda }$ considered by Owa [15].

Definition 1. 

The fractional integral of order λ is defined, for a function ϕ, by

$$D_{\zeta }^{-\lambda }\varphi \left(\zeta \right)={\displaystyle \frac{1}{\Gamma \left(\lambda \right)}}\underset{0}{\overset{\zeta }{\int }}{\displaystyle \frac{\varphi \left(\zeta \right)}{{(\zeta -u)}^{1-\lambda }}}du(\lambda <0),$$

(5)

where $\varphi \left(\zeta \right)$ is an analytic function in a simply connected region of the z-plane containing the origin, and the multiplicity of ${(\zeta -u)}^{\lambda -1}$ is removed by requiring $log(\zeta -u)$ to be real when $\zeta -u>0.$

Definition 2. 

The fractional derivative of order λ is defined, for a function $\varphi ,$ by

$$D_{\zeta }^{\lambda }\varphi \left(\zeta \right)={\displaystyle \frac{1}{\Gamma (1-\lambda )}}{\displaystyle \frac{d}{d\zeta }}\underset{0}{\overset{\zeta }{\int }}{\displaystyle \frac{\varphi \left(\zeta \right)}{{(\zeta -u)}^{\lambda }}}du(0\le \lambda <1),$$

(6)

where ϕ is an analytic function in a simply connected region of the z-plane containing the origin, and the multiplicity of ${(\zeta -\zeta )}^{-\lambda }$ is removed, as in Definition 1.

Definition 3. 

Under the hypotheses of Definition 2, the fractional derivative of order $n+\lambda$ is defined, for a function, $\varphi ,$ by

$$D_{\zeta }^{n+\lambda }\varphi \left(\zeta \right)={\displaystyle \frac{d^n}{d{\zeta }^n}}{D}_{\zeta }^{\lambda }\varphi \left(\zeta \right)(0\le \lambda <1;n\in {\mathbf{N}}_0).$$

(7)

By using these definitions of fractional calculus, Srivastava and Owa [16] defined the linear operator ${\Omega }^{\lambda }:\mathcal{A}\to \mathcal{A}$ by

$${\Omega }^{\lambda }\varphi \left(\zeta \right)=\Gamma (2-\lambda ){\zeta }^{\lambda }{D}_{\zeta }^{\lambda }\varphi \left(\zeta \right)(\lambda \ne 2,3,4,\dots ;\varphi \in \mathcal{A}).$$

(8)

Then, it is easily observed that

$${\Omega }^{\lambda }\varphi =\mathcal{L}(2,2-\lambda )\varphi .$$

(9)

We see that the Carlson–Shaffer operator plays an important role in the space of harmonic functions. We extend it symmetrically in the space of harmonic functions. Harmonic functions are important in the theory of minimal surfaces and also in various problems of applied mathematics (see, for example [13,17,18,19]). In 1984, Clunie and Sheil-Small [20] initiated the study of the geometric properties of harmonic functions. We say that complex-valued function $\varphi$ is harmonic in $\Delta$ if it has continuous second order-partial derivatives, which satisfies the Laplace equation

$$\Delta \varphi :={\displaystyle \frac{{\partial }^2\varphi }{\partial x^2}}+{\displaystyle \frac{{\partial }^2\varphi }{\partial y^2}}=0.$$

We note that every function, $\varphi$, harmonic in $\Delta$ can be uniquely represented as

$$\varphi =h+\overline{g},$$

(10)

where $h,g$ are analytic functions in $\Delta$ with $g\left(0\right)=0$. It is well known that a mapping, $\varphi$, is locally univalent and sense-preserving in $\Delta$ if and only if the Jacobian $J_{\varphi }$ of $\varphi$ is positive in $\Delta$. Lewy [13] proved that the converse is true for harmonic mappings. Since

$$J_{\varphi }\left(\zeta \right)={\left|{\displaystyle \frac{\partial \varphi }{\partial \zeta }}\right|}^2-{\left|{\displaystyle \frac{\partial \varphi }{\partial \overline{\zeta }}}\right|}^2\left(\zeta \in \Delta \right),$$

we obtain that $\varphi$ is sense-preserving and locally univalent if and only if

$$\left|h^{\prime }\left(\zeta \right)\right|>\left|g^{\prime }\left(\zeta \right)\right|\left(\zeta \in \Delta \right).$$

(11)

With $\mathcal{H}$, we denote the class of harmonic functions that are sense-preserving and univalent in $\Delta$. Moreover, with ${\mathcal{H}}_{\upsilon }$ we denote the class of function $\varphi \in \mathcal{H}$ of the form

$$\varphi \left(\zeta \right)={\eta }_1\zeta +\sum _{k=2}^{\infty }\left({\eta }_k{\zeta }^k+\overline{{\theta }_k{\zeta }^k}\right)\left({\eta }_1>0,\zeta \in \Delta \right),$$

(12)

with fixed point $\upsilon \in \left(0,1\right),$ i.e.,

$$\varphi \left(\upsilon \right)=\upsilon .$$

(13)

Now, we define Carlson and Shaffer’s extended ${\mathcal{L}}_{\tau }(\eta ,\vartheta ):\mathcal{H}\to \mathcal{H}$ by the formula

$${\mathcal{L}}_{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right):=\mathcal{L}(\eta ,\vartheta )h\left(\zeta \right)+\overline{\tau \mathcal{L}(\eta ,\vartheta )g\left(\zeta \right)}\left(\varphi =h+\overline{g}\in \mathcal{H}\right),$$

(14)

where $\tau =\pm 1.$

In particular, we have

$${\mathcal{L}}_1{(1;1)\varphi =\varphi ,\mathcal{L}}_{-1}(2;1)\varphi \left(\zeta \right){=\zeta h}^{\prime }\left(\zeta \right)-\overline{\zeta g^{\prime }\left(\zeta \right)}=:J_{\mathcal{H}}\varphi \left(\zeta \right).$$

It can be observed that, when $g\equiv 0$ in (10), the class $\mathcal{H}$ reduces to the well-known class $\mathcal{A}$ of analytic functions. Now, we denote with ${\mathcal{S}}_{\mathcal{H}}$ a subclass of ${\mathcal{H}}_0$ that contains univalent functions. The class ${\mathcal{S}}_{\mathcal{H}}$, with some of its geometric properties, was studied for the first time in [9]. A domain, $\mathbb{D}$ in $\mathbb{C}$, is said to be starlike with respect to the point $w_0$ if the line segment joining $w_0$ to any other point $\zeta$ in $\mathbb{D}$ remains in $\mathbb{D}$. Then, the set $\mathbb{D}$ is known as starlike. In particular, if $w_0=0$ and $\varphi (\Delta \left(r\right))$ is starlike, then a function, $\varphi \in {\mathcal{S}}_{\mathcal{H}}$ is considered as harmonic starlike in $\Delta \left(r\right)$. It is easy to verify, that a function $\varphi \in {\mathcal{H}}_{\rho }$ isharmonic-starlike in $\Delta \left(r\right)$ if and only if

$${\displaystyle \frac{\partial }{\partial t}}\left(arg\varphi \left(r{e}^{it}\right)\right)\ge 0\left(0\le t\le 2\pi \right)$$

or, equivalently,

$$\mathrm{Re}{\displaystyle \frac{J_{\mathcal{H}}\varphi \left(\zeta \right)}{\varphi \left(\zeta \right)}}\ge 0\left(\left|\zeta \right|=r\right).$$

(15)

Harmonic mapping was the most attractive problem of complex analysis in the middle of the 1980s. Many of the traditional results for conformal mappings are also valid for harmonic mappings, according to Clunie and Sheil-Small. This subject had progressively evolved at that time, but a vast number of problems remained unsolved.

In this proposed paper, by using the Carlson–Shaffer operator, we consider some generalization of the class of harmonic starlike functions with two fixed points, $\zeta =0$ and $\zeta =\upsilon$. To define the main class of harmonic functions, we need a definition of the weak subordination due to Mauir [21].

Definition 4. 

A function, $\phi :\Delta \to \mathbb{C}$, is said to be weakly subordinate to a function, $\Phi :\Delta \to \mathbb{C}$, if $\phi \left(\mathbb{U}\right)\subset \Phi \left(\mathbb{U}\right).$ Then, we write $\varphi \left(\zeta \right)\prec F\left(\zeta \right)$ (or simply $\varphi \prec F$).

Let $\eta ,\vartheta ,\Theta ,\Psi$ be real numbers; $\eta >0,\vartheta >1,-1\le -\Psi \le \Theta <\Psi <1$.

Definition 5. 

With ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$, we denote the class of functions $\varphi \in {\mathcal{H}}_{\upsilon }$ of the following form:

$$\varphi \left(\zeta \right)={\eta }_1\zeta -\sum _{k=2}^{\infty }\left(\left|{\eta }_k\right|{\zeta }^k-\tau \left|{\theta }_k\right|\overline{{\zeta }^k}\right)\left(\zeta \in \Delta \right),$$

(16)

which also satisfies the condition

$${\displaystyle \frac{J_{\mathcal{H}}{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)}{{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)}}\prec {\displaystyle \frac{1+\Theta \zeta }{1+\Psi \zeta }}.$$

(17)

In particular, we obtain the class ${\mathcal{Q}}_{\upsilon }^m(\Theta ,\Psi ):={\mathcal{Q}}_{\upsilon }^{{\left(-1\right)}^m}(m+1,1;\Theta ,\Psi )$ related to the Ruscheweyh derivatives [11] (see also [1,2,5,8,22]). The classes ${\mathcal{Q}}_{\mathcal{H}}^{*}\left(\alpha \right):={\mathcal{Q}}_{\upsilon }^0(2\alpha -1,1),$ ${\mathcal{Q}}_{\mathcal{H}}^c\left(\alpha \right):={\mathcal{Q}}_{\upsilon }^1(2\alpha -1,1)$ are the classes of starlike and convex functions of order $\alpha ,$ studied by Jahangiri [23]. And finally, the class ${\mathcal{Q}}_{\mathcal{H}}^{*}:={\mathcal{Q}}_{\upsilon }^{*}\left(0\right)$ consists of functions that are starlike in each disk $\Delta \left(\rho \right),$ $\rho \in \left(0,1\right.〉$, and the class ${\mathcal{Q}}_{\mathcal{H}}^c:={\mathcal{Q}}_{\upsilon }^c\left(0\right)$ consists of functions $\varphi \in \mathcal{H}$ that are convex in each disk $\Delta \left(\rho \right),$ $\rho \in \left(0,1\right.〉.$

In the paper, we obtain some coefficient criteria, distortion theorem, radii of starlikeness and convexity, and extreme points in the defined class of functions.

2. Coefficients Criteria

We start with the result, which will be basic in our investigations.

Theorem 1. 

Let $\varphi \in {\mathcal{H}}_{\upsilon }$ be of the form (16). Then, $\varphi \in$${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ if and only if

$$\sum _{k=2}^{\infty }\left(\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|{\eta }_k\right|+\left({\beta }_k+\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right)\le 1,$$

(18)

where

$$\begin{array}{c}\hfill {\alpha }_k:={\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)\left(\Psi -\Theta \right)}}\left\{k\left(1+\Psi \right)-\left(1+\Theta \right)\right\},\\ \hfill {\beta }_k:={\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)\left(\Psi -\Theta \right)}}\left\{k\left(1+\Psi \right)+\left(1+\Theta \right)\right\}.\end{array}$$

(19)

Proof. 

$\left(\Leftarrow \right)$ Let $\varphi \in {\mathcal{H}}_{\upsilon }$ of the form (16) satisfies (18). Then, $\varphi \in {\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ if and only if there exists a, function $\omega :\Delta \to \Delta$, such that

$${\displaystyle \frac{J_{\mathcal{H}}{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)}{{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)}}={\displaystyle \frac{1+\Theta \omega \left(\zeta \right)}{1+\Psi \omega \left(\zeta \right)}}\left(\zeta \in \Delta \right),$$

or, equivalently,

$$\left|{\displaystyle \frac{J_{\mathcal{H}}{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)-{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)}{\Psi J_{\mathcal{H}}{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)-\Theta {\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)}}\right|<1\left(\zeta \in \Delta \right).$$

(20)

Since $\varphi \left(\upsilon \right)=\upsilon ,$ we have

$${\eta }_1=1+\sum _{k=2}^{\infty }\left(\left|{\eta }_k\right|-\tau \left|{\theta }_k\right|\right){\upsilon }^{k-1}.$$

(21)

Thus, for $\left|\zeta \right|=\rho <1,$ we obtain

$$\left|J_{\mathcal{H}}{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)-{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)\right|-\left|\Psi J_{\mathcal{H}}{\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)-\Theta {\mathcal{L}}^{\tau }(\eta ,\vartheta )\varphi \left(\zeta \right)\right|$$

$$\begin{array}{c}{\displaystyle =\left|\sum _{k=2}^{\infty }\frac{(\eta ,k-1)}{(\vartheta ,k-1)}\left\{\left(k-1\right)\left|{\eta }_k\right|{\zeta }^k-\left(k+1\right)\left|{\theta }_k\right|{\overline{\zeta }}^k\right\}\right|}\hfill \\ -\left|\left(\Psi -\Theta \right){\eta }_1\zeta +\sum _{k=2}^{\infty }{\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)}}\left\{\left(\Psi k-\Theta \right)\left|{\eta }_k\right|{\zeta }^k-\left(\Psi k+\Theta \right)\left|{\theta }_k\right|{\overline{\zeta }}^k\right\}\right|\hfill \\ \le \sum _{k=2}^{\infty }{\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)}}\left\{\left(k-1\right)\left|{\eta }_k\right|+\left(k+1\right)\left|{\theta }_k\right|\right\}{\rho }^k-\left(\Psi -\Theta \right){\eta }_1\rho \hfill \\ +\sum _{k=2}^{\infty }{\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)}}\left\{\left(\Psi k-\Theta \right)\left|{\eta }_k\right|+\left(\Psi k+\Theta \right)\left|{\theta }_k\right|\right\}{\rho }^k\hfill \\ =\rho \sum _{k=2}^{\infty }{\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)}}\left\{\left[k\left(1+\Psi \right)-\left(1+\Theta \right)\right]\left|{\eta }_k\right|+\left[k\left(1+\Psi \right)+\left(1+\Theta \right)\right]\left|{\theta }_k\right|\right\}{\rho }^{k-1}\hfill \\ -\left(\Psi -\Theta \right)\rho -\left(\Psi -\Theta \right)\rho \sum _{k=2}^{\infty }\left(\left|{\eta }_k\right|-\tau \left|{\theta }_k\right|\right){\upsilon }^{k-1}\hfill \\ =\left(\Psi -\Theta \right)\rho \left\{\sum _{k=2}^{\infty }\left\{\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|{\eta }_k\right|+\left({\beta }_k+\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right\}{\rho }^{k-1}-1\right\}<0.\hfill \end{array}$$

Thus, we get (20).

$\left(\Rightarrow \right)$ Now, let $\varphi \in {\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ).$ Then, it satisfies (20) and $\varphi \left(\upsilon \right)=\upsilon .$ Thus, with (16), we can write

$$\left|{\displaystyle \frac{{\displaystyle \sum _{k=2}^{\infty }}\frac{(\eta ,k-1)}{(\vartheta ,k-1)}\left\{\left(k-1\right)\left|{\eta }_k\right|{\zeta }^k+\left(k+1\right)\left|{\theta }_k\right|{\overline{\zeta }}^k\right\}}{\left(\Psi -\Theta \right){\eta }_1\zeta -{\displaystyle \sum _{k=2}^{\infty }}\frac{(\eta ,k-1)}{(\vartheta ,k-1)}\left\{\left(\Psi k-\Theta \right)\left|{\eta }_k\right|{\zeta }^k+\left(\Psi k+\Theta \right)\left|{\theta }_k\right|{\overline{\zeta }}^k\right\}}}\right|<1(\zeta \in \Delta ).$$

Therefore, for $\zeta =\rho \in [0,1)$, we get

$${\displaystyle \frac{{\displaystyle \sum _{k=2}^{\infty }}\frac{(\eta ,k-1)}{(\vartheta ,k-1)}\left\{\left(k-1\right)\left|{\eta }_k\right|+\left(k+1\right)\left|{\theta }_k\right|\right\}{\rho }^{k-1}}{\left(\Psi -\Theta \right){\eta }_1-{\displaystyle \sum _{k=2}^{\infty }}\frac{(\eta ,k-1)}{(\vartheta ,k-1)}\left\{\left(\Psi k-\Theta \right)\left|{\eta }_k\right|+\left(\Psi k+\Theta \right)\left|{\theta }_k\right|\right\}{\rho }^{k-1}}}<1.$$

(22)

The denominator

$$d\left(\rho \right):=\left(\Psi -\Theta \right){\eta }_1-\sum _{k=2}^{\infty }{\displaystyle \frac{(\eta ,k-1)}{(\vartheta ,k-1)}}\left\{\left(\Psi k-\Theta \right)\left|{\eta }_k\right|+\left(\Psi k+\Theta \right)\left|{\theta }_k\right|\right\}{\rho }^{k-1}$$

is the continuous function that cannot vanish in $\left.〈0,1\right).$ Moreover, $d\left(0\right)>0,$ and, in consequence, d is positive in $\left.〈0,1\right).$ Therefore, according to (22) we get

$$\sum _{k=2}^{\infty }\left({\alpha }_k\left|{\eta }_k\right|+{\beta }_k\left|{\theta }_k\right|\right){\rho }^{k-1}<{\eta }_1.$$

Moreover, using (21), we obtain

$$\sum _{k=2}^{\infty }\left(\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|{\eta }_k\right|+\left({\beta }_k+\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right){\rho }^{k-1}<1(0\le \rho <1),$$

which yields assertion (18). □

Example 1. 

The functions $h_k$ and $g_k$ of the form

$$h_k\left(\zeta \right)={\displaystyle \frac{{\alpha }_k\zeta -{\zeta }^k}{{\alpha }_k-{\upsilon }^{k-1}}},g_k\left(\zeta \right)={\displaystyle \frac{{\beta }_k\zeta +\tau {\overline{\zeta }}^k}{{\beta }_k+\tau {\upsilon }^{k-1}}}(\zeta \in \Delta ;k\in {\mathbb{N}}_2)$$

(23)

satisfy conditions (13) and (18). Thus, they belong to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$.

With Theorem 1, we get the following two results.

Corollary 1. 

Let be a function, ϕ, of the form (16) that belongs to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$, and let $\left({\alpha }_k\right),\left({\beta }_k\right)$ be defined through (19). If

$${\alpha }_k-{\upsilon }^{k-1}>0,{\beta }_k+\tau {\upsilon }^{k-1}>0\left(k\in {\mathbb{N}}_2\right),$$

(24)

then,

$$\left|{\eta }_k\right|\le {\displaystyle \frac{1}{{\alpha }_k-{\upsilon }^{k-1}}},\left|{\theta }_k\right|\le {\displaystyle \frac{1}{{\beta }_k+\tau {\upsilon }^{k-1}}}\left(k\in {\mathbb{N}}_2\right).$$

(25)

The result is sharp; the functions $h_k$ and $g_k$ of the form (23) are the extremal functions.

Proof. 

With (24), all of the components of the sum (18) are positive. Thus, we have the estimations (25). Moreover, the coefficients of the functions $h_k,g_k$ of the form (23) realize equality in the inequalities (18). Thus the estimations are sharp. □

Corollary 2. 

Let $\left({\alpha }_k\right),\left({\beta }_k\right)$ be defined by (19). If ${\alpha }_k-{\upsilon }^{k-1}=0,$ then the coefficient ${\eta }_k$ in the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ is unbounded. If ${\beta }_k+\tau {\upsilon }^{k-1}=0,$ then the coefficient ${\theta }_k$ in the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ is unbounded. Moreover, if there exists $k\in {\mathbb{N}}_2=\left\{2,3,\cdots \right\}$ such that

$${\alpha }_k-{\upsilon }^{k-1}<0\mathit{or}{\beta }_k+\tau {\upsilon }^{k-1}<0$$

then all of the coefficients in the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ are unbounded.

Proof. 

If

$${\alpha }_{k_0}-{\upsilon }^{k_0-1}\le 0$$

for some $k_0\in {\mathbb{N}}_2,$ then the function

$${\varphi }_{k_0}\left(\zeta \right)=\left(1+\eta {\upsilon }^{k_0-1}\right)\zeta -\eta {\zeta }^{k_0}(\zeta \in \Delta )$$

is in the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ for $\eta >0.$ Thus, the coefficient ${\eta }_{k_0}=-\eta$ is unbounded. If there exists $k_0\in {\mathbb{N}}_2$ such that

$${\alpha }_{k_0}-{\upsilon }^{k_0-1}<0,$$

(26)

then, for any $k\in {\mathbb{N}}_2$ such that

$${\alpha }_k-{\upsilon }^{k-1}>0,$$

(27)

the function

$${\varphi }_k\left(\zeta \right)=\left(1+\eta {\upsilon }^{k_0-1}+\theta {\zeta }^{k-1}\right)\zeta -\eta {\zeta }^{k_0}-\theta {\zeta }^k,$$

where

$$\theta ={\displaystyle \frac{1+\left({\upsilon }^{k_0-1}-{\alpha }_{k_0}\right)\eta }{{\alpha }_k-{\upsilon }^{k-1}}},$$

belong to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ).$ Since $\theta$ can be any positive real number, the coefficient ${\eta }_k=\theta$ is unbounded. Analogously, if

$${\beta }_{k_0}+\tau {\upsilon }^{k_0-1}\le 0$$

for some $k_0\in {\mathbb{N}}_2,$ then the function

$${\varphi }_{k_0}\left(\zeta \right)=\left(1-\tau \eta {\upsilon }^{k_0-1}\right)\zeta +\tau \eta \overline{{\zeta }^{k_0}}(\zeta \in \Delta )$$

belongs to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ for all positive real numbers $\eta .$ Thus the coefficient ${\theta }_{k_0}=\eta$ is unbounded. If there exists an integer, $k_0\in {\mathbb{N}}_2$, such that

$${\beta }_{k_0}+\tau {\upsilon }^{k_0-1}<0$$

then, for any $k\in {\mathbb{N}}_2$ such that

$${\beta }_k+\tau {\upsilon }^{k-1}>0,$$

the function

$${\varphi }_k\left(\zeta \right)=\left(1-\tau \eta {\upsilon }^{k_0-1}-\tau \theta {\upsilon }^{k-1}\right)\zeta +\tau \eta {\overline{\zeta }}^{k_0}+\tau \theta {\overline{\zeta }}^k,$$

where

$$\theta ={\displaystyle \frac{1-\left({\beta }_{k_0}+\tau {\upsilon }^{k_0-1}\right)\eta }{{\beta }_k+\tau {\upsilon }^{k-1}}},$$

belong to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ).$ Since $\theta$ can be any positive real number, the coefficient ${\theta }_k=\theta$ is unbounded, and the proof is completed. □

3. Dispersion Theorems

Through Theorem 1, we get the following result.

Lemma 1. 

Let $\varphi \in {\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ be of the form (16), and let $\left({\alpha }_k\right),$ $\left({\beta }_k\right)$ be defined by (19). If

$$0<{\alpha }_2-\upsilon \le {\alpha }_k-{\upsilon }^{k-1},0<{\beta }_2+\tau \upsilon \le {\beta }_k+\tau {\upsilon }^{k-1}\left(k\in {\mathbb{N}}_2\right),$$

(28)

then

$$\sum _{k=2}^{\infty }\left|{\eta }_k\right|\le {\displaystyle \frac{1}{{\alpha }_2-\upsilon }},\sum _{k=2}^{\infty }\left|{\theta }_k\right|\le {\displaystyle \frac{1}{{\beta }_2+\tau \upsilon }}.$$

Moreover, if

$$0<{\displaystyle \frac{{\alpha }_2-\upsilon }{2}}\le {\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k}},0<{\displaystyle \frac{{\beta }_2+\tau \upsilon }{2}}\le {\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{2}}\left(k\in {\mathbb{N}}_2\right),$$

(29)

then

$$\sum _{k=2}^{\infty }k\left|{\eta }_k\right|\le {\displaystyle \frac{2}{{\alpha }_2-\upsilon }},\sum _{k=2}^{\infty }k\left|{\theta }_k\right|\le {\displaystyle \frac{2}{{\beta }_2+\tau \upsilon }}.$$

Remark 1. 

The second part of Lemma 1 can be written in terms of the σ-neighborhood $N_{\sigma }$ defined by

$$N_{\sigma }=\left\{\varphi \left(\zeta \right)={\eta }_1\zeta +\sum _{k=2}^{\infty }\left({\eta }_k{\zeta }^k+\overline{{\theta }_k{\zeta }^k}\right)\in {\mathcal{H}}_{\upsilon }:\sum _{k=2}^{\infty }\left(\left|{\eta }_k\right|+\left|{\theta }_k\right|\right)\le \sigma \right\}$$

in the following form:

$${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )\subset N_{\sigma },$$

where

$$\delta ={\displaystyle \frac{2}{{\alpha }_2-\upsilon }}+{\displaystyle \frac{2}{{\beta }_2+\tau \upsilon }}.$$

Theorem 2. 

Let $\varphi \in$ ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ),$ $\left|\zeta \right|=\rho <1.$ If the sequences $\left({\alpha }_k\right),$ $\left({\beta }_k\right)$ given by (19) satisfy inequalities (28), then

$${\varphi }_1\left(\rho \right)\le \left|\varphi \left(\zeta \right)\right|\le {\varphi }_2\left(\rho \right),$$

(30)

where

$$\begin{array}{ccc}\hfill {\varphi }_1\left(\rho \right)& :& =\left\{\begin{array}{cc}\rho & \left(\tau =-1,\rho \le \upsilon \right)\\ {\displaystyle \frac{{\beta }_2\rho -{\rho }^2}{{\beta }_2+\upsilon }}& \left(\tau =1,\rho \le \upsilon \right)\\ {\displaystyle \frac{\left({\alpha }_2{\beta }_2+\tau {\upsilon }^2\right)\rho -\left({\alpha }_2+{\beta }_2-\left(1-\tau \right)\upsilon \right){\rho }^2}{\left({\alpha }_2-\upsilon \right)\left({\beta }_2-\upsilon \right)}}& \left(\rho >\tau \upsilon \right)\end{array}\right.,\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill {\varphi }_2\left(\rho \right)& :& =\left\{\begin{array}{cc}{\displaystyle \frac{{\alpha }_2\rho +{\rho }^2}{{\alpha }_2-\upsilon }}& \left(\rho \le \tau \upsilon \right)\\ {\displaystyle \frac{\left({\alpha }_2{\beta }_2+\tau {\upsilon }^2\right)\rho +\left({\alpha }_2+{\beta }_2-\left(1-\tau \right)\upsilon \right){\rho }^2}{\left({\alpha }_2-\upsilon \right)\left({\beta }_2-\upsilon \right)}}& \left(\rho >\tau \upsilon \right)\end{array}\right..\hfill \end{array}$$

(32)

The result is sharp, with the extremal functions $h_k,g_k$ of the form (23) and the functions $h_1,$ ${\varphi }_{\tau }$ of the form

$$h_1\left(\zeta \right)=\zeta ,{\varphi }_{\tau }\left(\zeta \right)={\displaystyle \frac{{\alpha }_2{\beta }_2+\tau {\upsilon }^2}{\left({\alpha }_2-\upsilon \right)\left({\beta }_2+\tau \upsilon \right)}}\zeta -{\displaystyle \frac{{\zeta }^2}{{\alpha }_2-\upsilon }}+{\displaystyle \frac{\tau {\overline{\zeta }}^2}{{\beta }_2+\tau \upsilon }}(\zeta \in \Delta ).$$

(33)

Proof. 

Suppose that the function $\varphi$ of the form (16) belongs to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$. Through Lemma 1, we have

$$\sum _{k=2}^{\infty }\left|{\eta }_k\right|\le {\displaystyle \frac{1}{{\alpha }_2-\upsilon }},\sum _{k=2}^{\infty }\left|{\theta }_k\right|\le {\displaystyle \frac{1}{{\beta }_2+\tau \upsilon }}.$$

First, we observe, that the sequence $\left\{\left({\rho }^{k-1}+{\upsilon }^{k-1}\right)\right\}$ is decreasing and positive. Also, $\left\{\left({\rho }^{k-1}-{\upsilon }^{k-1}\right)\right\}$ is the positive decreasing sequence for $\rho >\upsilon$. In consequence, we get

$$\begin{array}{c}0<\rho +\upsilon \le {\rho }^{k-1}+{\upsilon }^{k-1}\left(\rho ,\upsilon \in \left(0,1\right),k\in {\mathbb{N}}_2\right),\hfill \\ 0<\rho -\upsilon \le {\rho }^{k-1}-{\upsilon }^{k-1}\left(0<\upsilon <\rho <1,k\in {\mathbb{N}}_2\right).\hfill \end{array}$$

(34)

Moreover, we have

$$\begin{array}{cc}\left|\varphi \right(\zeta \left)\right|\hfill & =\left|{\eta }_1\zeta -{\displaystyle \sum _{k=2}^{\infty }}\left({\eta }_k{\zeta }^k-\overline{\tau {\theta }_k{\zeta }^k}\right)\right|\le \rho \left({\eta }_1+{\displaystyle \sum _{k=2}^{\infty }}\left(\left|{\eta }_k\right|+\left|{\theta }_k\right|\right){\rho }^{k-1}\right)\hfill \\ \hfill & \le \rho \left(1+{\displaystyle \sum _{k=2}^{\infty }}\left(\left|{\eta }_k\right|-\tau \left|{\theta }_k\right|\right){\upsilon }^{k-1}+{\displaystyle \sum _{k=2}^{\infty }}\left(\left|{\eta }_k\right|+\left|{\theta }_k\right|\right){\rho }^{k-1}\right)\hfill \\ \hfill & \le \rho \left(1+{\displaystyle \sum _{k=2}^{\infty }}({\rho }^{k-1}+{\upsilon }^{k-1})\left|{\eta }_k\right|+{\displaystyle \sum _{k=2}^{\infty }}\left({\rho }^{k-1}-\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right).\hfill \end{array}$$

If $\rho >\tau \upsilon ,$ then through (34), we have

$$\begin{array}{ccc}\hfill \left|\varphi \left(\zeta \right)\right|& \le & \rho \left(1+(\rho +\upsilon )\sum _{k=2}^{\infty }\left|{\eta }_k\right|+\left(\rho -\tau \upsilon \right)\sum _{k=2}^{\infty }\left|{\theta }_k\right|\right)\hfill \\ & \le & \rho \left(1+{\displaystyle \frac{\rho +\upsilon }{{\alpha }_2-\upsilon }}+{\displaystyle \frac{\rho -\tau \upsilon }{{\beta }_2+\tau \upsilon }}\right)={\displaystyle \frac{\left({\alpha }_2{\beta }_2+\tau {\upsilon }^2\right)\rho +\left({\alpha }_2+{\beta }_2-\left(1-\tau \right)\upsilon \right){\rho }^2}{\left({\alpha }_2-\upsilon \right)\left({\beta }_2-\upsilon \right)}}.\hfill \end{array}$$

If $\rho \le \tau \upsilon ,$ then, through (34), we have

$$\begin{array}{ccc}\hfill \left|\varphi \left(\zeta \right)\right|& \le & \rho \left(1+\sum _{k=2}^{\infty }({\rho }^{k-1}+{\upsilon }^{k-1})\left|{\eta }_k\right|\right)\hfill \\ & \le & \rho \left(1+(\upsilon +\rho )\sum _{k=2}^{\infty }\left|{\eta }_k\right|\right)\le {\displaystyle \frac{{\alpha }_2\rho +{\rho }^2}{{\alpha }_2-\upsilon }}.\hfill \end{array}$$

Analogously, we get

$$\begin{array}{cc}\left|\varphi \right(\zeta \left)\right|\hfill & =\left|{\eta }_1\zeta -{\displaystyle \sum _{k=2}^{\infty }}\left({\eta }_k{\zeta }^k-\overline{\tau {\theta }_k{\zeta }^k}\right)\right|\ge \rho \left({\eta }_1-{\displaystyle \sum _{k=2}^{\infty }}\left(\left|{\eta }_k\right|+\left|{\theta }_k\right|\right){\rho }^{k-1}\right)\hfill \\ \hfill & =\rho \left(1+{\displaystyle \sum _{k=2}^{\infty }}\left(\left|{\eta }_k\right|-\tau \left|{\theta }_k\right|\right){\upsilon }^{k-1}-{\displaystyle \sum _{k=2}^{\infty }}\left(\left|{\eta }_k\right|+\left|{\theta }_k\right|\right){\rho }^{k-1}\right)\hfill \\ \hfill & =\rho \left(1-{\displaystyle \sum _{k=2}^{\infty }}({\rho }^{k-1}-{\upsilon }^{k-1})\left|{\eta }_k\right|-{\displaystyle \sum _{k=2}^{\infty }}\left({\rho }^{k-1}+\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right).\hfill \end{array}$$

If $\tau =-1$ and $\rho \le \upsilon ,$ then we obtain $\left|\varphi \left(\zeta \right)\right|\geqq \rho .$ If $\tau =1$ and $\rho \le \upsilon ,$ then, via (34), we have

$$\begin{array}{ccc}\hfill \left|\varphi \left(\zeta \right)\right|& \ge & \rho \left(1-\sum _{k=2}^{\infty }\left({\rho }^{k-1}+{\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right)\ge \rho \left(1-(\rho +\upsilon )\sum _{k=2}^{\infty }\left|{\theta }_k\right|\right)\hfill \\ & \ge & \rho \left(1-{\displaystyle \frac{\rho +\upsilon }{{\beta }_2+\upsilon }}\right)={\displaystyle \frac{{\beta }_2\rho -{\rho }^2}{{\beta }_2+\upsilon }}.\hfill \end{array}$$

If $\rho >\upsilon ,$ then, via (34), we obtain

$$\begin{array}{ccc}\hfill \left|\varphi \left(\zeta \right)\right|& \ge & \rho \left(1-(\rho -\upsilon )\sum _{k=2}^{\infty }\left|{\eta }_k\right|-(\rho +\tau \upsilon )\sum _{k=2}^{\infty }\left|{\theta }_k\right|\right)\hfill \\ & \ge & \rho \left(1-{\displaystyle \frac{\rho -\upsilon }{{\alpha }_2-\upsilon }}-{\displaystyle \frac{\rho +\tau \upsilon }{{\beta }_2+\tau \upsilon }}\right)={\displaystyle \frac{\left({\alpha }_2{\beta }_2+\tau {\upsilon }^2\right)\rho -\left({\alpha }_2+{\beta }_2-\left(1-\tau \right)\upsilon \right){\rho }^2}{\left({\alpha }_2-\upsilon \right)\left({\beta }_2-\upsilon \right)}},\hfill \end{array}$$

and we have the complete assertion (30). □

If $\eta \ge \vartheta >0,$ then the sequences $\left({\alpha }_k\right),$ $\left({\beta }_k\right)$ defined by (19) satisfy (28). Thus, through Theorem 2, we have the following corollary.

Corollary 3. 

Let a function, ϕ belong to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ),$ and let $\left|\zeta \right|=\rho <1.$ If $\eta \ge \vartheta >0,$ and then the estimation (30) holds true. The result is sharp, with the extremal functions of the forms (23) and (33).

Moreover, through Corollary 2 we have the following complementary result.

Corollary 4. 

Let $\left({\alpha }_k\right),\left({\beta }_k\right)$ be defined by (19). If there exists $k\in {\mathbb{N}}_2$ such that

$${\alpha }_k-{\upsilon }^{k-1}\le 0\mathrm{or}{\beta }_k+\tau {\upsilon }^{k-1}\le 0,$$

then the sets

$${\Theta }_{\rho }=\left\{\left|\varphi \left(\zeta \right)\right|:\varphi \in {\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ),\left|\zeta \right|=\rho \right\}$$

are unbounded for each $\rho \in \left(0,1\right)$.

4. The Radii of Starlikeness and Convexity

A function, $\varphi \in \mathcal{H}$, is said to be harmonic-starlike in $\Delta \left(\rho \right)$ if $\varphi \left(\Delta \left(\rho \right)\right)$ is a starlike domain with respect to the point $w_0=0$ (see [23,24]).

Lemma 2. 

A function, $\varphi \in {\mathcal{H}}_{\upsilon }$, of the form (16) is starlike in $\Delta \left(\rho \right)$ if and only if it satisfies the condition

$$\sum _{k=2}^{\infty }\left(\left(k-{\upsilon }^{k-1}\right)\left|{\eta }_k\right|+\left(k+\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right){\rho }^{k-1}\le 1.$$

(35)

Proof. 

If $\varphi \in {\mathcal{H}}_{\upsilon }$ is starlike in $\Delta \left(\rho \right),$ then the curve $\varphi \left(\partial \Delta \left(\rho \right)\right)$ is starlike with respect to the point $w_0=0$; i.e.,

$${\displaystyle \frac{\partial }{\partial \varsigma }}\left(arg\varphi \left(\rho e^{i\varsigma }\right)\right)>0\left(\varsigma \in \left[0,2\pi \right]\right).$$

(36)

It is easy to verify that the condition (36) can be written as

$$\mathrm{Re}{\displaystyle \frac{J_{\mathcal{H}}\varphi \left(\zeta \right)}{\varphi \left(\zeta \right)}}>0(\zeta \in \Delta \left(\rho \right)),$$

or, equivalently,

$$\left|{\displaystyle \frac{J_{\mathcal{H}}\varphi \left(\zeta \right)-\varphi \left(\zeta \right)}{J_{\mathcal{H}}\varphi \left(\zeta \right)+\varphi \left(\zeta \right)}}\right|<1(\zeta \in \Delta \left(\rho \right)).$$

(37)

Since, for $\left|\zeta \right|=\rho <1$, we have

$$\begin{array}{cc}\hfill \left|{\displaystyle \frac{J_{\mathcal{H}}\varphi \left(\zeta \right)-\varphi \left(\zeta \right)}{J_{\mathcal{H}}\varphi \left(\zeta \right)+\varphi \left(\zeta \right)}}\right|& =\left|{\displaystyle \frac{{\displaystyle \sum _{k=2}^{\infty }}\left((k-1)\left|{\eta }_k\right|{\zeta }^k-(k+1)\left|{\theta }_k\right|{\overline{\zeta }}^k\right)}{2{\eta }_1\zeta +{\displaystyle \sum _{k=2}^{\infty }}\left((k+1)\left|{\eta }_k\right|{\zeta }^k-(k-1)\left|{\theta }_k\right|{\overline{\zeta }}^k\right)}}\right|\hfill \\ \hfill & \le {\displaystyle \frac{{\displaystyle \sum _{k=2}^{\infty }}\left((k-1)\left|{\eta }_k\right|+\left(k+1\right)\left|{\theta }_k\right|\right){\rho }^{k-1}}{2{\eta }_1-{\displaystyle \sum _{k=2}^{\infty }}\left((k+1)\left|{\eta }_k\right|+(k-1)\left|{\theta }_k\right|\right){\rho }^{k-1}}}.\hfill \end{array}$$

condition (37) is true if and only if

$$\sum _{k=2}^{\infty }k\left(\left|{\eta }_k\right|+\left|{\theta }_k\right|\right){\rho }^{k-1}\le {\eta }_1,.$$

(38)

Since $\varphi \left(\upsilon \right)=\upsilon ,$ i.e.,

$${\eta }_1=1+\sum _{k=2}^{\infty }\left(\left|{\eta }_k\right|-\tau \left|{\theta }_k\right|\right){\upsilon }^{k-1},$$

(39)

the condition (38) is equivalent to (35).

□

Analogously, a function, $\varphi \in \mathcal{H}$, is said to be harmonic–convex in $\Delta \left(\rho \right)$ if $\varphi \left(\Delta \left(\rho \right)\right)$ is a convex domain, i.e.,

$${\displaystyle \frac{\partial }{\partial \varsigma }}\left(arg{\displaystyle \frac{\partial }{\partial \varsigma }}\varphi \left(\rho e^{i\varsigma }\right)\right)>0\left(\varsigma \in \left[0,2\pi \right]\right).$$

It is clear, that any function convex in $\Delta \left(\rho \right)$ is also starlike in $\Delta \left(\rho \right).$ Moreover, we have the following equivalence.

Definition 6. 

We say that the number $R^{*}\left(\varphi \right)\in \left[0,1\right]$ is the radius of starlikeness of the function $\varphi \in \mathcal{H}$ if ϕ is starlike in $\Delta \left(\rho \right)$ for all $\rho \le R^{*}\left(\varphi \right).$ Similarly, we say that the number $R^c\left(\varphi \right)\in \left[0,1\right]$ is the radius of convexity of the function $\varphi \in \mathcal{H}$ if ϕ is convex in $\Delta \left(\rho \right)$ for all $\rho \le R^c\left(\varphi \right).$

Definition 7. 

We say that the number $R^{*}\left(\ominus \right)\in \left[0,1\right]$ is the radius of starlikeness of the class $\ominus \subset \mathcal{H}$ if each function, $\varphi \in \ominus$, is starlike in $\Delta \left(\rho \right)$, and we say that the number $R^c\left(\ominus \right)\in \left[0,1\right]$ is the radius of convexity of the class ⊖ if each function, $\varphi \in \ominus$, is convex in $\Delta \left(\rho \right)$.

From the definitions, we have

$$R_{\alpha }^{*}(\ominus ):=inf\left\{R^{*}\left(\varphi \right):\varphi \in \ominus \right\},R^c(\ominus ):=inf\left\{R^c\left(\varphi \right):\varphi \in \ominus \right\}.$$

Corollary 5. 

Let $\ominus \subset {\mathcal{H}}_{\upsilon }$. If $R^{*}\left(\ominus \right)>0,$ then all of the coefficient of the class ⊖ are bounded; i.e., for each $k\in {\mathbb{N}}_2$ there exists $M_k>0$, such that

$$\left|{\eta }_k\right|\le M_k,\left|{\theta }_k\right|\le M_k\left(\varphi \in \ominus \right),$$

(40)

where ϕ is of the form (16).

Proof. 

Let $\varphi \in {\mathcal{H}}_{\upsilon }$ be of the form (16) and $\rho :=R^{*}\left(\ominus \right)>0$. Then, through Lemma 2, we have

$$\begin{array}{ccc}\hfill \left|{\eta }_k\right|& \le & {\displaystyle \frac{1}{\left(k-{\upsilon }^{k-1}\right){\rho }^{k-1}}}<{\displaystyle \frac{1}{{\rho }^{k-1}}}=:M_k,\hfill \\ \hfill \left|{\theta }_k\right|& \le & {\displaystyle \frac{1}{\left(k+\tau {\upsilon }^{k-1}\right){\rho }^{k-1}}}<{\displaystyle \frac{1}{{\rho }^{k-1}}}=M_k.\hfill \end{array}$$

Thus, all of the coefficients of the class $\ominus$ are bounded. □

Corollary 6. 

A function, $\varphi \in \mathcal{H}$, is convex in $\Delta \left(\rho \right)$ if and only if the function $g=J_{\mathcal{H}}$ϕ is starlike in $\Delta \left(\rho \right).$

Corollary 6 can be written in the following form.

Corollary 7. 

If there exists an unbounded coefficient of the class $\ominus \subset {\mathcal{H}}_{\upsilon }$, then $R^{*}\left(\ominus \right)=0$.

Thus, via Corollary 2, we have the following corollary.

Corollary 8. 

Let $\left({\alpha }_k\right),\left({\beta }_k\right)$ be defined by (19). If there exists $k\in {\mathbb{N}}_2$ such that

$${\alpha }_k-{\upsilon }^{k-1}\le 0\mathit{or}{\beta }_k+\tau {\upsilon }^{k-1}\le 0,$$

then

$$R^{*}\left({\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )\right)=0.$$

Theorem 3. 

Let $\left({\alpha }_k\right),\left({\beta }_k\right)$ be defined by (19). If

$${\alpha }_k-{\upsilon }^{k-1}>0,{\beta }_k+\tau {\upsilon }^{k-1}>0\left(k\in {\mathbb{N}}_2\right),$$

then

$$R^{*}\left({\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )\right)=\underset{k\in {\mathbb{N}}_2}{inf}{\left(min\left\{{\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k-{\upsilon }^{k-1}}},{\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{k+\tau {\upsilon }^{k-1}}}\right\}\right)}^{{\displaystyle \frac{1}{k-1}}}.$$

(41)

Proof. 

Let $\varphi \in {\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$ be defined by (16) with (13). Via Theorem 1, we have

$$\sum _{k=2}^{\infty }\left(\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|{\eta }_k\right|+\left({\beta }_k+\tau {\upsilon }^{k-1}\right)\left|{\theta }_k\right|\right)\le 1,$$

Thus, the condition (35) is true if

$$\left(k-{\upsilon }^{k-1}\right){\rho }^{k-1}\le {\alpha }_k-{\upsilon }^{k-1},\left(k+\tau {\upsilon }^{k-1}\right){\rho }^{k-1}\le {\beta }_k+\tau {\upsilon }^{k-1}\left(k\in {\mathbb{N}}_2\right),$$

or, equivalently,

$$\rho \le {\left(min\left\{{\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k-{\upsilon }^{k-1}}},{\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{k+\tau {\upsilon }^{k-1}}}\right\}\right)}^{{\displaystyle \frac{1}{k-1}}}\left(k\in {\mathbb{N}}_2\right).$$

(42)

It follows that $R^{*}\left(\varphi \right)\ge {\rho }^{*}$, where

$${\rho }^{*}:=\underset{k\in {\mathbb{N}}_2}{inf}{\left(min\left\{{\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k-{\upsilon }^{k-1}}},{\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{k+\tau {\upsilon }^{k-1}}}\right\}\right)}^{{\displaystyle \frac{1}{k-1}}},$$

and, in consequence ${R^{*}({\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )\ge \rho }^{*}.$ Moreover, for the functions $h_k,g_k$ defined by (23), we get

$$R^{*}\left(h_k\right)={\left({\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k-{\upsilon }^{k-1}}}\right)}^{{\displaystyle \frac{1}{k-1}}},R^{*}\left(g_k\right)={\left({\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{k+\tau {\upsilon }^{k-1}}}\right)}^{{\displaystyle \frac{1}{k-1}}}.$$

Thus, the radius $R^{*}\left({\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )\right)$ cannot be larger than ${\rho }^{*}$, and wehave (41). □

From Theorem 3 and Corollary 6, we have the following result.

Theorem 4. 

Let $\left({\alpha }_k\right),\left({\beta }_k\right)$ be defined by (19). If

$${\alpha }_k-{\upsilon }^{k-1}>0,{\beta }_k+\tau {\upsilon }^{k-1}>0\left(k\in {\mathbb{N}}_2\right),$$

then

$$R^c\left({\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )\right)=\underset{k\in {\mathbb{N}}_2}{inf}{\left({\displaystyle \frac{1}{k}}min\left\{{\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k-{\upsilon }^{k-1}}},{\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{k+\tau {\upsilon }^{k-1}}}\right\}\right)}^{{\displaystyle \frac{1}{k-1}}}.$$

(43)

Let

$$u_k:=\left({\displaystyle \frac{{\alpha }_k-{\upsilon }^{k-1}}{k-{\upsilon }^{k-1}}}\right),v_k:=\left({\displaystyle \frac{{\beta }_k+\tau {\upsilon }^{k-1}}{k+\tau {\upsilon }^{k-1}}}\right).$$

Then, for $\eta \ge \vartheta$ we have

$$u_k>1,v_k>1\left(k\in {\mathbb{N}}_2\right).$$

Moreover, if $\eta \ge \vartheta +1,$ then

$${\displaystyle \frac{u_k}{k}}>1,{\displaystyle \frac{v_k}{k}}>1\left(k\in {\mathbb{N}}_2\right).$$

Thus, through Theorems 3 and 4, we have the following two corollaries.

Corollary 9. 

If $\eta \ge \vartheta ,$ then $R^{*}($ ${\mathcal{Q}}_{\upsilon }^{\tau }($η,ϑ;Θ,Ψ$\left)\right)=1.$ This means, that ${\mathcal{Q}}_{\upsilon }^{\tau }($η,ϑ;Θ,Ψ$)\subset$${\mathcal{Q}}_{\upsilon }^{*}.$

Corollary 10. 

If $\eta \ge \vartheta +1,$ then $R^c$(${\mathcal{Q}}_{\upsilon }^{\tau }($η,ϑ;Θ,Ψ$\left)\right)=1.$ This means, that ${\mathcal{Q}}_{\upsilon }^{\tau }($η,ϑ;Θ,Ψ$)\subset$${\mathcal{Q}}_{\upsilon }^{\vartheta }.$

5. Convexity and Extreme Points

Theorem 5. 

Let $h_1\left(\varsigma \right)=\varsigma$, let $h_k,g_k$ be defined by (23), and let $\left({\alpha }_k\right),\left({\beta }_k\right)$, defined by (19), satisfy (24)

$${\alpha }_k-{\upsilon }^{k-1}>0,{\beta }_k-\tau {\upsilon }^{k-1}>0\left(k\in {\mathbb{N}}_2\right)$$

A function, ϕ, belongs to the class ${\mathcal{Q}}_{\upsilon }^{\tau }($η,ϑ;Θ,Ψ) if and only if

$$\varphi ={\lambda }_1{h}_1+\sum _{k=2}^{\infty }\left({\lambda }_k{h}_k+{\mu }_k{g}_k\right),$$

(44)

where

$${\lambda }_1+\sum _{k=2}^{\infty }\left({{\lambda }_k+\mu }_k\right)={1and{\lambda }_k,\mu }_k\ge 0.$$

(45)

Proof. 

$\left(\Rightarrow \right)$ Let $\varphi$ of the form (16) belong to the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ).$ If we put

$${\lambda }_k:=\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|a_k\right|\ge 0,{\mu }_k:=\left({\beta }_k-\tau {\upsilon }^{k-1}\right)\left|b_k\right|\ge 0$$

and

$${\lambda }_1:=1-\sum _{k=2}^{\infty }\left({{\lambda }_k+\mu }_k\right),$$

then, through (18), we have ${\lambda }_1\ge 0,$ i.e., (45) holds. Moreover, we have

$$h_k\left(\varsigma \right)={\displaystyle \frac{{\alpha }_k\varsigma -{\varsigma }^k}{{\alpha }_k-{\upsilon }^{k-1}}},g_k\left(\varsigma \right)={\displaystyle \frac{{\beta }_k\varsigma +\tau {\overline{\varsigma }}^k}{{\beta }_k-\tau {\upsilon }^{k-1}}}(\varsigma \in \ngeqq ;k\in {\mathbb{N}}_2)$$

(46)

$$\begin{array}{ccc}& & \left({\lambda }_1{h}_1+\sum _{k=2}^{\infty }\left({\lambda }_k{h}_k+{\mu }_k{g}_k\right)\right)\left(\varsigma \right)=\left(1-\sum _{k=2}^{\infty }\left({{\lambda }_k+\mu }_k\right)\right)\varsigma \hfill \\ & & +\sum _{k=2}^{\infty }\left(\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|a_k\right|{\displaystyle \frac{{\alpha }_k\varsigma -{\varsigma }^k}{{\alpha }_k-{\upsilon }^{k-1}}}+\left({\beta }_k-\tau {\upsilon }^{k-1}\right)\left|b_k\right|{\displaystyle \frac{{\beta }_k\varsigma +\tau {\overline{\varsigma }}^k}{{\beta }_k-\tau {\upsilon }^{k-1}}}\right)\hfill \\ & =& \left(1-\sum _{k=2}^{\infty }\left(\left({\alpha }_k-{\upsilon }^{k-1}\right)|a_k|+\left({\beta }_k-\tau {\upsilon }^{k-1}\right)\left|b_k\right|\right)\right)\varsigma \hfill \\ & & +\varsigma \sum _{k=2}^{\infty }\left({\alpha }_k\left|a_k\right|+{\beta }_k\left|b_k\right|\right)-\sum _{k=2}^{\infty }\left(|a_k|{\varsigma }^k-\tau \left|b_k\right|{\overline{\varsigma }}^k\right)\hfill \\ & =& \left(1+\sum _{k=2}^{\infty }\left({\upsilon }^{k-1}\left|a_k\right|+\tau {\upsilon }^{k-1}\left|b_k\right|\right)\right)\varsigma -\sum _{k=2}^{\infty }\left(|a_k|{\varsigma }^k-\tau \left|b_k\right|{\overline{\varsigma }}^k\right)=\varphi \left(\varsigma \right).\hfill \end{array}$$

and the condition (44) follows.

$(\Leftarrow )$. Now, let a function, $\varphi$, of the form (16) satisfy (44). Thus,

$$\begin{array}{ccc}\hfill \varphi \left(\varsigma \right)& =& {\lambda }_1{h}_1\left(\varsigma \right)+\sum _{k=2}^{\infty }\left({\lambda }_k{h}_k+{\mu }_k{g}_k\right)\left(\varsigma \right)\hfill \\ & =& \left(1-\sum _{k=2}^{\infty }\left({{\lambda }_k+\mu }_k\right)\right)\varsigma +\sum _{k=2}^{\infty }\left({\lambda }_k{\displaystyle \frac{{\alpha }_k\varsigma -{\varsigma }^k}{{\alpha }_k-{\upsilon }^{k-1}}}+{\mu }_k{\displaystyle \frac{{\beta }_k\varsigma +\tau {\overline{\varsigma }}^k}{{\beta }_k-\tau {\upsilon }^{k-1}}}\right)\hfill \\ & =& a_1\varsigma -\sum _{k=2}^{\infty }\left(\left|a_k\right|{\varsigma }^k-\tau \left|b_k\right|\overline{{\varsigma }^k}\right),\hfill \end{array}$$

where

$$\left|a_k\right|={\displaystyle \frac{{\lambda }_k}{{\alpha }_k-{\upsilon }^{k-1}}},\left|b_k\right|={\displaystyle \frac{\tau {\mu }_k}{{\beta }_k+\tau {\upsilon }^{k-1}}},$$

and

$$a_1=1+\sum _{k=2}^{\infty }\left(\left|a_k\right|+\tau \left|b_k\right|\right){\upsilon }^{k-1}.$$

Thus, the function $\varphi$ is of the form (16), and

$$\sum _{k=2}^{\infty }\left(\left({\alpha }_k-{\upsilon }^{k-1}\right)\left|a_k\right|+\left({\beta }_k+\tau {\upsilon }^{k-1}\right)\left|b_k\right|\right)=\sum _{k=2}^{\infty }\left({\lambda }_k+{\mu }_k\right)\le 1.$$

Finally, we have $\varphi$∈${\mathcal{Q}}_{\upsilon }^{\tau }($ $\eta$,$\vartheta$;$\Theta$,$\Psi$), which ends the proof. □

Let $\mathcal{F}$ be a subclass of theclass $\mathcal{H}$. A functions, $\varphi \in \mathcal{F}$, iscalled an extreme point of $\mathcal{F}$ if the condition

$$\varphi =\gamma {\varphi }_1+\left(1-\gamma \right){\varphi }_2\left({\varphi }_1,{\varphi }_2\in \mathcal{F},0<\gamma <1\right)$$

implies ${\varphi }_1={\varphi }_2=\varphi .$ We shall use the notation $E\mathcal{F}$ to denote the set of all extreme points of $\mathcal{F}.$

We say that aclass $\mathcal{F}$ is convex if

$$\gamma \varphi +(1-\gamma )\psi \in \mathcal{F}(\varphi ,\psi \in \mathcal{F},0\le \gamma \le 1).$$

From Theorem 5, we get the following corollary.

Corollary 11. 

The class ${\mathcal{Q}}_{\upsilon }^{\tau }($η,ϑ;Θ,Ψ) is convex. Moreover,

$$E{\mathcal{Q}}_{\upsilon }^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )=\left\{h_k:k\in \mathbb{N}\right\}\cup \left\{g_k:k\in {\mathbb{N}}_2\right\},$$

where $h_1\left(\varsigma \right)=\varsigma$ and $h_k,g_k$ are the functions of the form (23).

6. Conclusions and Declarations

By using the Carlson–Shaffer operator, we have considered the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta \vartheta ;\Theta ,\Psi )$ of harmonic functions with fixed two points. By choosing the parameters of the defined class of functions, we can obtain several new and well-known results for harmonic functions (see, for example [1,2,5,8,22,23,24,25,26,27,28,29]). Some results restricted to analytic functions are obtained in [10,30,31]. We can notice here that functions from the class ${\mathcal{Q}}_{\upsilon }^{\tau }(\eta \vartheta ;\Theta ,\Psi )$ have the normalization $\varphi \left(0\right)=0,$ $\varphi \left(\upsilon \right)=\upsilon$ or, equivalently,

$$\varphi \left(0\right)=0,{\displaystyle \frac{\varphi \left(\upsilon \right)-\varphi \left(0\right)}{\upsilon -0}}=1.$$

Thus, for $\upsilon \to 0^{+}$ we can get the classical normalization $\varphi \left(0\right)=0,$ ${\varphi }_{\zeta }^{\prime }\left(0\right)=1.$ Therefore, putting $\upsilon =0$ in the obtained results, we get related results for the class

$${\mathcal{Q}}^{\tau }(\eta ,\vartheta ;\Theta ,\Psi ):={\mathcal{Q}}_0^{\tau }(\eta ,\vartheta ;\Theta ,\Psi )$$

of functions with the classical normalization.