Properties of Spiral-Like Close-to-Convex Functions Associated with Conic Domains

Abstract

In this paper, our aim is to define certain new classes of multivalently spiral-like, starlike, convex and the varied Mocanu-type functions, which are associated with conic domains. We investigate such interesting properties of each of these function classes, such as (for example) sufficiency criteria, inclusion results and integral-preserving properties.

1. Introduction and Motivation

Let $\mathcal{A}\left(p\right)$ denote the class of functions of the form:

$$f\left(z\right)=z^p+\sum _{n=1}^{\infty }{a}_{n+p}{z}^{n+p}(p\in \mathbb{N}=\{1,2,3,\cdots \}),$$

(1)

which are analytic and p-valent in the open unit disk:

$$\mathbb{E}=\{z:z\in \mathbb{C}\mathrm{and}\left|z\right|<1\}.$$

In particular, we write:

$$\mathcal{A}\left(1\right)=\mathcal{A}.$$

Furthermore, by $\mathcal{S}\subset \mathcal{A}$, we shall denote the class of all functions that are univalent in $\mathbb{E}.$

The familiar class of p-valently starlike functions in $\mathbb{E}$ will be denoted by ${\mathcal{S}}^{*}\left(p\right)$, which consists of functions $f\in \mathcal{A}\left(p\right)$ that satisfy the following conditions:

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>0(\forall z\in \mathbb{E}).$$

One can easily see that:

$${\mathcal{S}}^{*}\left(1\right)={\mathcal{S}}^{*},$$

where ${\mathcal{S}}^{*}$ is the well-known class of normalized starlike functions (see [1]).

We denote by $\mathcal{K}$ the class of close-to-convex functions, which consists of functions $f\in \mathcal{A}$ that satisfy the following inequality:

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)}{g\left(z\right)}\right)>0(\forall z\in \mathbb{E})$$

for some $g\in {\mathcal{S}}^{*}.$

For two functions f and g analytic in $\mathbb{E}$, we say that the function f is subordinate to the function g and write as follows:

$$f\prec g\mathrm{or}f\left(z\right)\prec g\left(z\right),$$

if there exists a Schwarz function w, which is analytic in $\mathbb{E}$ with:

$$w\left(0\right)=0\mathrm{and}\left|w\left(z\right)\right|<1,$$

such that:

$$f\left(z\right)=g\left(w\left(z\right)\right).$$

Furthermore, if the function g is univalent in $\mathbb{E}$, then it follows that:

$$f\left(z\right)\prec g\left(z\right)(z\in \mathbb{E})⟹f\left(0\right)=g\left(0\right)\mathrm{and}f\left(\mathbb{E}\right)\subset g\left(\mathbb{E}\right).$$

Next, for a function $f\in \mathcal{A}\left(p\right)$ given by (1) and another function $g\in \mathcal{A}\left(p\right)$ given by:

$$g\left(z\right)=z^p+\sum _{n=2}^{\infty }{b}_{n+p}{z}^{n+p}\left(\forall z\in \mathbb{E}\right),$$

the convolution (or the Hadamard product) of f and g is given by:

$$\left(f*g\right)\left(z\right)=z^p+\sum _{n=2}^{\infty }{a}_{n+p}{b}_{n+p}{z}^{n+p}=\left(g*f\right)\left(z\right).$$

The subclass of $\mathcal{A}$ consisting of all analytic functions with a positive real part in $\mathbb{E}$ is denoted by $\mathcal{P}$. An analytic description of $\mathcal{P}$ is given by:

$$h\left(z\right)=1+\sum _{n=1}^{\infty }{c}_n{z}^n(\forall z\in \mathbb{E}).$$

Furthermore, if:

$$\Re \left\{h\left(z\right)\right\}>\rho ,$$

then we say that h is in the class $\mathcal{P}\left(\rho \right).$ Clearly, one see that:

$$\mathcal{P}\left(0\right)=\mathcal{P}.$$

Historically, in the year 1933, Spaček [2] introduced the $\beta$-spiral-like functions as follows.

Definition 1.

A function $f\in \mathcal{A}$ is said to be in the class ${\mathcal{S}}^{*}\left(\beta \right)$ if and only if:

$$\Re \left(e^{i\beta }\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>0(\forall z\in \mathbb{E})$$

for:

$$\beta \in \mathbb{R}\mathit{and}\left|\beta \right|<\frac{\pi }{2},$$

where $\mathbb{R}$ is the set of real numbers.

In the year 1967, Libera [3] extended this definition to the class of functions, which are spiral-like of order $\rho$ denoted by ${\mathcal{S}}_{\rho }^{*}\left(\beta \right)$ as follows.

Definition 2.

A function $f\in \mathcal{A}$ is said to be in the class ${\mathcal{S}}_{\rho }^{*}\left(\beta \right)$ if and only if:

$$\Re \left(e^{i\beta }\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>\rho (\forall z\in \mathbb{E})$$

$$\left(0\leqq \rho <1;\beta \in \mathbb{R}\mathit{and}\left|\beta \right|<\frac{\pi }{2}\right),$$

where $\mathbb{R}$ is the set of real numbers.

The above function classes ${\mathcal{S}}^{*}\left(\beta \right)$ and ${\mathcal{S}}_{\rho }^{*}\left(\beta \right)$ have been studied and generalized by different viewpoints and perspectives. For example, in the year 1974, a subclass $S_{\beta }^{\alpha }\left(\rho \right)$ of spiral-like functions was introduced by Silvia (see [4]), who gave some remarkable properties of this function class. Subsequently, Umarani [5] defined and studied another function class $SC(\alpha ,\beta )$ of spiral-like functions. Recently, Noor et al. [6] generalized the works of Silvia [4] and Umarani [5] by defining the class $M(p,\alpha ,\beta ,\rho )$. Here, in this paper, we define certain new subclasses of spiral-like close-to-convex functions by using the idea of Noor et al. [6] and Umarani [5].

We now recall that Kanas et al. (see [7,8]; see also [9]) defined the conic domains ${\Omega }_k(k\geqq 0)$ as follows:

$${\Omega }_k=\left\{u+iv:u>k\sqrt{{\left(u-1\right)}^2+v^2}\right\}.$$

(2)

By using these conic domains ${\Omega }_k(k\geqq 0)$, they also introduced and studied the corresponding class k-$\mathcal{ST}$ of k-starlike functions (see Definition 3 below).

Moreover, for fixed $k$, ${\Omega }_k$ represents the conic region bounded successively by the imaginary axis for $\left(k=0\right),$ for $k=1$ a parabola, for $0<k<1$ the right branch of a hyperbola, and for $k>1$ an ellipse. For these conic regions, the following functions $p_k\left(z\right),$ which are given by (3), play the role of extremal functions.

$$p_k\left(z\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1+z}{1-z}}=1+2z+2z^2+\cdots \hfill & \left(k=0\right)\hfill \\ 1+{\displaystyle \frac{2}{{\pi }^2}}{\left(log{\displaystyle \frac{1+\sqrt{z}}{1-\sqrt{z}}}\right)}^2\hfill & \left(k=1\right)\hfill \\ 1+{\displaystyle \frac{2}{1-k^2}}{sinh}^2\left\{\left({\displaystyle \frac{2}{\pi }}arccosk\right)arctan\left(h\sqrt{z}\right)\right\}\hfill & \left(0\leqq k<1\right)\hfill \\ 1+{\displaystyle \frac{1}{k^2-1}}sin\left({\displaystyle \frac{\pi }{2K\left(\kappa \right)}}{\int }_0^{{\displaystyle \frac{u\left(z\right)}{\sqrt{\kappa }}}}{\displaystyle \frac{dt}{\sqrt{1-t^2}\sqrt{1-{\kappa }^2{t}^2}}}\right)+{\displaystyle \frac{1}{k^2-1}}\hfill & \left(k>1\right),\hfill \end{array}\right.$$

(3)

where:

$$u\left(z\right)=\frac{z-\sqrt{\kappa }}{1-\sqrt{\kappa }z}\left(\forall z\in \mathbb{E}\right)$$

and $\kappa \in (0,1)$ is chosen such that:

$$k=cosh\left(\frac{\pi K^{\prime }\left(\kappa \right)}{4K\left(\kappa \right)}\right).$$

Here, $K\left(\kappa \right)$ is Legendre’s complete elliptic integral of the first kind and:

$$K^{\prime }\left(\kappa \right)=K\left(\sqrt{1-{\kappa }^2}\right),$$

that is, $K^{\prime }\left(\kappa \right)$ is the complementary integral of $K\left(\kappa \right)$.

These conic regions are being studied and generalized by several authors (see, for example, [10,11,12,13]).

The class k-$\mathcal{ST}$ is defined as follows.

Definition 3.

A function $f\in \mathcal{A}$ is said to be in the class k-$\mathcal{ST}$ if and only if:

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec p_k\left(z\right)\left(\forall z\in \mathbb{E};k\geqq 0\right)$$

or, equivalently,

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)>k\left|\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}-1\right|.$$

The class of k-uniformly close-to-convex functions denoted by k-$\mathcal{UK}$ was studied by Acu [14].

Definition 4.

A function $f\in \mathcal{A}$ is said to be in the class k-$\mathcal{UK}$ if and only if:

$$\Re \left(\frac{z{f}^{\prime }\left(z\right)}{g\left(z\right)}\right)>k\left|\frac{z{f}^{\prime }\left(z\right)}{g\left(z\right)}-1\right|,$$

where $g\in k$-$\mathcal{ST}$.

In recent years, several interesting subclasses of analytic functions were introduced and investigated from different viewpoints (see, for example, [6,15,16,17,18,19,20]; see also [21,22,23,24,25]). Motivated and inspired by the recent and current research in the above-mentioned work, we here introduce and investigate certain new subclasses of analytic and p-valent functions by using the concept of conic domains and spiral-like functions as follows.

Definition 5.

Let $f\in \mathcal{A}\left(p\right).$ Then, $f\in k$-$\mathcal{K}(p,\lambda )$ for a real number λ with $\left|\lambda \right|<\frac{\pi }{2}$ if and only if:

$$\Re \left(\frac{e^{i\lambda }}{p}\frac{z{f}^{\prime }\left(z\right)}{\psi \left(z\right)}\right)>k\left|\frac{z{f}^{\prime }\left(z\right)}{\psi \left(z\right)}-p\right|+\rho cos\lambda \left(k\geqq 0;0\leqq \rho <1\right)$$

for some $\psi \in {\mathcal{S}}^{*}$.

Definition 6.

Let $f\in \mathcal{A}\left(p\right)$. Then, $f\in k$-$\mathcal{Q}(p,\lambda )$ for a real λ with $\left|\lambda \right|<\frac{\pi }{2}$ if and only if:

$$\Re \left(\frac{e^{i\lambda }}{p}\frac{z{f}^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}\right)>k\left|\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}-p\right|+\rho cos\lambda \left(k\geqq 0;0\leqq \rho <1\right)$$

for some $\psi \in \mathcal{C}$.

Definition 7.

Let $f\in \mathcal{A}\left(p\right)$ with:

$$\frac{f^{\prime }\left(z\right)f\left(z\right)}{pz}\ne 0$$

and for some real ϕ and λ with $\left|\lambda \right|<\frac{\pi }{2}$. Then, $f\in k$-$\mathcal{Q}\left(\varphi ,\lambda ,\eta ,f,\psi \right)$ if and only if:

$$\Re \left(\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)\right)>k\left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|+\rho cos\lambda ,$$

where

$$\begin{array}{cc}\hfill \mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)& =(e^{i\lambda }-\varphi cos\lambda )\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}\hfill \\ & +\frac{\varphi cos\lambda }{p-\eta }\left(\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}-\eta \right)\left(\frac{-1}{2}\leqq \eta <1\right).\hfill \end{array}$$

(4)

2. A Set of Lemmas

Each of the following lemmas will be needed in our present investigation.

Lemma 1.

(see [26] p. 70) Let h be a convex function in $\mathbb{E}$ and:

$$q:\mathbb{E}⟹\mathbb{C}\mathit{and}\Re \left(q\left(z\right)\right)>0(z\in \mathbb{E}).$$

If p is analytic in $\mathbb{E}$ with:

$$p\left(0\right)=h\left(0\right),$$

then:

$$p\left(z\right)+q\left(z\right)z{p}^{\prime }\left(z\right)\prec h\left(z\right)\mathit{implies}p\left(z\right)\prec h\left(z\right).$$

Lemma 2.

(see [26] p. 195) Let h be a convex function in $\mathbb{E}$ with:

$$h\left(0\right)=0\mathit{and}A>1.$$

Suppose that $j\geqq \frac{4}{h^{\prime }\left(0\right)}$ and that the functions $B\left(z\right)$, $C\left(z\right)$ and $D\left(z\right)$ are analytic in $\mathbb{E}$ and satisfy the following inequalities:

$$\Re \left\{B\left(z\right)\right\}\geqq A+\left|C\left(z\right)-1\right|-\Re \left(C\left(z\right)-1\right)+jD\left(z\right),z\in \mathbb{E}.$$

If p is analytic in $\mathbb{E}$ with:

$$p\left(z\right)=1+a_{1}z+a_2{z}^2+\cdots$$

and the following subordination relation holds true:

$$A{z}^2{p}^{″}\left(z\right)+B\left(z\right)z{p}^{\prime }\left(z\right)+C\left(z\right)p\left(z\right)+D\left(z\right)\prec h\left(z\right),$$

then:

$$p\left(z\right)\prec h\left(z\right).$$

3. Main Results and Their Demonstrations

In this section, we will prove our main results.

Theorem 1.

A function $f\in \mathcal{A}$ is in the class k-$\mathcal{Q}\left(\varphi ,\lambda ,\eta ,f,\psi \right)$ if:

$$\sum _{n=2}^{\infty }{\ddot{U}}_n\left(p,\varphi ,\lambda ,\eta ,\xi \right)<p^2(p-\eta ),$$

where:

$$\begin{array}{cc}\hfill {\ddot{U}}_n\left(p,\varphi ,\lambda ,\eta ,\xi \right)& =\left(k+1\right)[(e^{i\lambda }-\varphi cos\lambda )(p-\eta )p+p^4\varphi cos\lambda \hfill \\ & +(e^{i\lambda }-\varphi cos\lambda )(p-\eta )(n+p)\left|a_{n+p}\right|+{(n+p)}^2\left|a_{n+p}\right|\hfill \\ & +[(np\varphi cos\lambda +p^3(p-\eta )](n+p)\left|b_{n+p}\right|+n{p}^2\varphi cos\lambda -p^3(p-\eta ).\hfill \end{array}$$

(5)

Proof. 

Let us assume that the relation (4) holds true. It now suffices to show that:

$$k\left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|-\Re \left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|<1.$$

(6)

We first consider:

$$\begin{array}{c}\left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|\hfill \\ =\left|\left(e^{i\lambda }-\varphi cos\lambda \right)\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}+\frac{\varphi cos\lambda }{\left(p-\eta \right)}\left(\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}-\eta \right)-p\right|\hfill \\ =\left|\frac{(e^{i\lambda }-\varphi cos\lambda )\left(p-\eta \right)f^{\prime }\left(z\right)}{p\left(p-\eta \right){\psi }^{\prime }\left(z\right)}+\frac{p\varphi cos\lambda {\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{p\left(p-\eta \right){\psi }^{\prime }\left(z\right)}-\right.\hfill \\ \left.-\frac{\eta p\varphi cos\lambda {\psi }^{\prime }\left(z\right)}{p\left(p-\eta \right){\psi }^{\prime }\left(z\right)}-\frac{p^2\left(p-\eta \right){\psi }^{\prime }\left(z\right)}{p(p-\eta ){\psi }^{\prime }\left(z\right)}\right|.\hfill \end{array}$$

Now, by using the series form of the functions f and $\psi$ given by:

$$f\left(z\right)=z^p+\sum _{n=2}^{\infty }{a}_{n+p}{z}^{n+p}$$

and:

$$\psi \left(z\right)=z^p+\sum _{n=2}^{\infty }{b}_{n+p}{z}^{n+p}$$

in the above relation, we have:

$$\begin{array}{c}\left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|\hfill \\ =\left|\frac{\left(e^{i\lambda }-\varphi cos\lambda \right)\left(p-\eta \right)\left(p{z}^{p-1}\right)+p\varphi cos\lambda \left(p^2{z}^{p-1}\right)}{p(p-\eta )\left(p{z}^{p-1}+{\sum }_{n=2}^{\infty }(n+p)b_{n+p}{z}^{n+p-1}\right)}\right.\hfill \\ \left.+\frac{{\sum }_{n=2}^{\infty }(n+p)a_{n+p}{z}^{n+p-1}[\left(e^{i\lambda }-\varphi cos\lambda \right)\left(p-\eta \right)+(n+p)]}{p\left(p-\eta \right)\left(p{z}^{p-1}+{\sum }_{n=2}^{\infty }(n+p)b_{n+p}{z}^{n+p-1}\right)}-\frac{n\varphi cos\lambda }{(p-\eta )}-p\right|\hfill \\ \leqq \frac{\left(e^{i\lambda }-\varphi cos\lambda \right)\left(p-\eta \right)\left(p\right)+p\varphi cos\lambda \left(p^2\right)}{p\left(p-\eta \right)\left(p+{\sum }_{n=2}^{\infty }(n+p)\left|b_{n+p}\right|\right)}\hfill \\ +\frac{{\sum }_{n=2}^{\infty }(n+p)\left|a_{n+p}\right|\left\{\left(e^{i\lambda }-\varphi cos\lambda \right)\left(p-\eta \right)+(n+p)\right\}}{p\left(p-\eta \right)\left(p+{\sum }_{n=2}^{\infty }(n+p)\left|b_{n+p}\right|\right)}-\left\{\frac{n\varphi cos\lambda }{(p-\eta )}+p\right\}.\hfill \end{array}$$

We now see that:

$$\begin{array}{c}k\left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|-\Re \left\{\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right\}\hfill \\ \leqq (k+1)\left|\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)-p\right|\hfill \\ \leqq \left(k+1\right)\left[\frac{\left(e^{i\lambda }-\varphi cos\lambda \right)\left(p-\eta \right)\left(p\right)+p\varphi cos\lambda \left(p^2\right)}{p\left(p-\eta \right)\left(p+{\sum }_{n=2}^{\infty }(n+p)\left|b_{n+p}\right|\right)}\right.\hfill \\ +\left.\frac{{\sum }_{n=2}^{\infty }(n+p)\left|a_{n+p}\right|[\left(e^{i\lambda }-\varphi cos\lambda \right)\left(p-\eta \right)+(n+p)]}{p\left(p-\eta \right)\left(p+{\sum }_{n=2}^{\infty }(n+p)\left|b_{n+p}\right|\right)}-\left[\frac{n\varphi cos\lambda }{(p-\eta )}+p\right]\right].\hfill \end{array}$$

The above inequality is bounded above by one, if:

$$\begin{array}{c}\left(k+1\right)\left[\left(e^{i\lambda }-\varphi cos\lambda \right)(p-\eta )p\right]+(p\varphi cos\lambda )p^2\hfill \\ +\left(\sum _{n=2}^{\infty }(n+p)\left|a_{n+p}\right|\right)\left\{(e^{i\lambda }-\varphi cos\lambda )(p-\eta )+(n+p)\right\}-\left[\frac{n\varphi cos\lambda }{(p-\eta )}-p\right]\hfill \\ ·\left\{p\left(p-\eta \right)\left(p+\sum _{n=2}^{\infty }(n+p)\left|b_{n+p}\right|\right)\right\}\hfill \\ \leqq p(p-\eta )p+\sum _{n=2}^{\infty }(n+p)\left|b_{n+p}\right|.\hfill \end{array}$$

Hence:

$$\sum _{n=2}^{\infty }{\ddot{U}}_n\left(p,\varphi ,\lambda ,\eta ,\xi \right)\leqq p^2(p-\eta ),$$

where ${\ddot{U}}_n\left(p,\varphi ,\lambda ,\eta ,\xi \right)$ is given by (5), which completes the proof of Theorem 1. □

Theorem 2.

A function $f\in \mathcal{A}\left(p\right)$ satisfies the condition:

$$\left|\frac{1}{e^{ij}F\left(z\right)}-\frac{1}{2\rho }\right|<\frac{1}{2\rho }\left(0\leqq \rho <1;j\in \mathbb{R}\right)$$

(7)

if and only if $f\in 0$-$\mathcal{K}(p,\lambda ),$ where

$$F\left(z\right)=\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}.$$

Proof. 

Suppose that f satisfies (7). We then can write:

$$\begin{array}{cc}\hfill \left|\frac{2\rho -e^{ij}F\left(z\right)}{e^{ij}F\left(z\right)2\rho }\right|& <\frac{1}{2\rho }\hfill \\ & ⟺{\left(\left|\frac{2\rho -e^{ij}F\left(z\right)}{e^{ij}F\left(z\right)2\rho }\right|\right)}^2<{\left(\frac{1}{2\rho }\right)}^2\hfill \\ & ⟺\left(2\rho -e^{ij}F\left(z\right)\right)\left(\overline{2\rho -e^{ij}F\left(z\right)}\right)<e^{-ij}\overline{F\left(z\right)}{e}^{ij}F\left(z\right)\hfill \\ & ⟺4{\rho }^2-2\rho \left[e^{-ij}\overline{F\left(z\right)}+e^{ij}F\left(z\right)\right]+F\left(z\right)\overline{F\left(z\right)}<F\left(z\right)\overline{F\left(z\right)}\hfill \\ & ⟺4{\rho }^2-2\rho \left[e^{-ij}\overline{F\left(z\right)}+e^{ij}F\left(z\right)\right]<0\hfill \\ & ⟺2\rho -2\Re \left[e^{ij}\overline{F\left(z\right)}\right]<0\hfill \\ & ⟺\Re \left[e^{ij}F\left(z\right)\right]>\rho \hfill \\ & ⟺\Re \left(e^{ij}\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}\right)>\rho .\hfill \end{array}$$

This completes the proof of Theorem 2. □

Theorem 3.

For $0\leqq {\phi }_1<{\phi }_2,$ it is asserted that:

$$k-\mathcal{Q}\left(p,{\phi }_2,\lambda ,\eta \right)\subset 0-\mathcal{Q}\left(p,{\phi }_1,\lambda ,\eta \right).$$

Proof. 

Let $f\left(z\right)\in k-\mathcal{Q}\left(p,{\phi }_2,\lambda ,\eta \right).$ Then:

$$\begin{array}{c}\frac{1}{p-\eta }\left[\left(e^{i\lambda }-{\varphi }_{1}cos\lambda \right)\left(p-\eta \right)\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}+{\phi }_{1}cos\lambda \left(\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{\psi {\left(z\right)}^{\prime }}-\eta \right)\right]\hfill \\ =\frac{{\phi }_1}{{\phi }_2}\left[\left(e^{i\lambda }-{\phi }_{2}cos\lambda \right)\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}+\frac{{\phi }_{2cos\lambda }}{\left(p-\eta \right)}\left(\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{p\psi {\left(z\right)}^{\prime }}-\eta \right)\right]\hfill \\ -\left(\frac{{\phi }_1-{\phi }_2}{{\phi }_2}\right)e^{i\lambda }\frac{s{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}\hfill \\ =\frac{{\phi }_1}{{\phi }_2}{H}_1\left(z\right)+\left(1-\frac{{\phi }_1}{{\phi }_2}\right)H_2\left(z\right)=H\left(z\right),\hfill \end{array}$$

where:

$$H_1\left(z\right)=\left(e^{i\lambda }-{\phi }_{2}cos\lambda \right)\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}+\frac{{\phi }_{2}cos\lambda }{\left(p-\eta \right)}\left(\frac{{\left(z{f}^{\prime }z\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}-\eta \right)\in \mathcal{P}\left(h_{k,\rho }\right)\subset \mathcal{P}\left(\rho \right)$$

and:

$$H_2\left(z\right)=e^{i\lambda }\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}\in \mathcal{P}\left(\rho \right).$$

Since $\mathcal{P}\left(\rho \right)$ is a convex set (see [27]), we therefore have $H\left(z\right)\in \mathcal{P}\left(\rho \right)$. This implies that $f\in 0-\mathcal{Q}\left(p,{\phi }_1,\lambda ,\eta \right)$. Thus:

$$k-\mathcal{Q}\left(p,{\phi }_2,\lambda ,\eta \right)\subset 0-\mathcal{Q}\left(p,{\phi }_1,\lambda ,\eta \right).$$

The proof of Theorem 3 is now completed. □

Theorem 4.

Let $\varphi >0$ and $\lambda <\frac{\pi }{2}$. Then:

$$k-\mathcal{Q}(p,\varphi ,\lambda ,\eta ,\xi )\subset k-\mathcal{K}(p,0,\xi ).$$

Proof. 

Let $f\in k$-$\mathcal{Q}(p,\varphi ,\lambda ,\eta ,\xi )$, and suppose that:

$$\frac{f^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}=p\left(z\right),$$

(8)

where $p\left(z\right)$ is analytic and $p\left(0\right)=1$. Now, by differentiating both sides of (8) with respect to z, we have:

$$\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}=z{p}^{\prime }\left(z\right)+p\left(z\right)\epsilon \left(z\right),$$

(9)

where:

$$\epsilon \left(z\right)=\frac{{\left(z{\psi }^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}.$$

By using (8) and (9) in (4), we arrive at:

$$\begin{array}{cc}\hfill \mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)& =\left(e^{i\lambda }-\varphi cos\lambda \right)\frac{p\left(z\right)}{p}+\frac{\varphi cos\lambda }{p-\eta }\left(z{p}^{\prime }\left(z\right)+p\left(z\right)\epsilon \left(z\right)-\eta \right)\hfill \\ & =\frac{\varphi cos\lambda }{p-\eta }z{p}^{\prime }\left(z\right)+\left(\frac{e^{i\lambda }}{p}-\frac{\varphi cos\lambda }{p-\epsilon \left(z\right)}\right)\left(\frac{\varphi cos\lambda }{p-\eta }\right)p\left(z\right)-\frac{\eta \varphi cos\lambda }{p-\eta }\hfill \\ & =B\left(z\right)z{p}^{\prime }\left(z\right)+C\left(z\right)p\left(z\right)+D\left(z\right),\hfill \end{array}$$

(10)

where:

$$B\left(z\right)=\frac{\varphi cos\lambda }{p-\eta },$$

$$C\left(z\right)=\frac{e^{i\lambda }\left(p-\eta \right)-\varphi cos\lambda \left(p-\eta \right)+\varphi cos\lambda \epsilon \left(z\right)p}{p(p-\eta )}$$

and:

$$D\left(z\right)=\frac{\eta \varphi cos\lambda }{p-\eta }.$$

Now, since $f\in k$-$\mathcal{Q}(p,\varphi ,\lambda ,\eta ,\xi ),$ we have:

$$B\left(z\right)z{p}^{\prime }\left(z\right)+C\left(z\right)p\left(z\right)+D\left(z\right)\prec p_k\left(z\right),$$

(11)

which, upon replacing $p\left(z\right)$ by:

$$p_{*}\left(z\right)=p\left(z\right)-1,$$

and $p_k\left(z\right)$ by:

$$p_k^{*}\left(z\right)=p_k\left(z\right)-1,$$

shows that the above subordination in (11) becomes as follows:

$$B\left(z\right)z{p}_x^{\prime }\left(z\right)+C\left(z\right)p_x\left(z\right)+D_{*}\left(z\right)\prec p_k^{*}\left(z\right),$$

(12)

where:

$$D_{*}\left(z\right)=C\left(z\right)+D\left(z\right)-1.$$

We now apply Lemma 2 with:

$$A=0$$

and

$$p_{*}\left(z\right)\prec p_k^{*}\left(z\right).$$

We thus find that:

$$\frac{f^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}=p\left(z\right)\prec p_k^{*}\left(z\right).$$

(13)

This complete the proof of Theorem 4. □

For $f\in \mathcal{A},$ we next consider the integral operator defined by:

$$F\left(z\right)=I_m\left[f\right]=\frac{m+1}{z^m}{\int }_0^z{t}^{m-1}f\left(t\right)dt.$$

(14)

This operator was given by Bernardi [28] in the year 1969. In particular, the operator $I_1$ was considered by Libera [29]. We prove the following result.

Theorem 5.

Let $f\left(z\right)\in k$-$\mathcal{Q}\left(p,\varphi ,\lambda ,\eta ,\xi \right).$ Then, $I_m\left[f\right]\in \mathcal{K}\left(p,0,\xi \right).$

Proof. 

Let the function $\psi \left(z\right)$ be such that:

$$\mathcal{M}\left(\varphi ,\lambda ,\eta ,f,\psi \right)=\left(e^{i\lambda }-\varphi cos\lambda \right)\frac{z{f}^{\prime }\left(z\right)}{p\psi \left(z\right)}+\frac{\varphi cos\lambda }{\left(p-\eta \right)}\left(\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}-\eta \right).$$

Then, according to [14], the function $G=I_m\left[f\right]\in \mathcal{CD}\left(k,\delta \right)$. Furthermore, from (14), we deduce that:

$$\left(1+m\right)f\left(z\right)=\left(1+m\right)F\left(z\right)+z{\left(F\left(z\right)\right)}^{\prime }$$

(15)

and:

$$\left(1+m\right)g\left(z\right)=\left(1+m\right)G\left(z\right)+z{\left(G\left(z\right)\right)}^{\prime }.$$

(16)

If we now put:

$$p\left(z\right)=\frac{F^{\prime }\left(z\right)}{G^{\prime }\left(z\right)}$$

and:

$$q\left(z\right)=\frac{1}{\left(m+1\right)+\left(\frac{z{G}^{″}\left(z\right)}{G^{\prime }\left(z\right)}\right)},$$

then, by simple computations, we find that:

$$\frac{f\left(z\right)}{\psi \left(z\right)}=\frac{\left(1+m\right)F^{\prime }\left(z\right)+z{F}^{″}\left(z\right)}{\left(1+m\right)G^{\prime }\left(z\right)+z{G}^{″}\left(z\right)}$$

or, equivalently, that:

$$\frac{f^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}=p\left(z\right)+z{p}^{\prime }\left(z\right)q\left(z\right).$$

(17)

We now let:

$$\frac{f^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}=p\left(z\right)+z{p}^{\prime }\left(z\right)q\left(z\right)=h\left(z\right),$$

(18)

where the function $h\left(z\right)$ is analytic in $\mathbb{E}$ with $h\left(0\right)=1.$ Then, by using (18), we have:

$$\frac{{\left(zf\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}=z{h}^{\prime }\left(z\right)+\epsilon \left(z\right)h\left(z\right),$$

(19)

where:

$$\epsilon \left(z\right)=\frac{{\left(z{\psi }^{\prime }\left(z\right)\right)}^{\prime }}{{\psi }^{\prime }\left(z\right)}.$$

Furthermore, by using (18) and (19) in (4), we obtain:

$$\begin{array}{cc}\hfill \mathcal{M}\left(\alpha ,\beta ,\gamma ,\lambda ,\delta ,f\right)& =\left(e^{i\lambda }-\theta cos\lambda \right)\frac{z{f}^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}+\frac{\varphi cos\lambda }{p-\eta }\left(\frac{{\left(z{f}^{\prime }\left(z\right)\right)}^{{}^{\prime }}}{{\psi }^{\prime }\left(z\right)}-\eta \right)\hfill \\ & =\left(e^{i\lambda }-\theta cos\lambda \right)+\frac{\varphi cos\lambda }{p-\eta }z{h}^{\prime }\left(z\right)+\left[z{h}^{\prime }\left(z\right)+\epsilon \left(z\right)h\left(z\right)-\eta \right]\hfill \\ & =\frac{\varphi cos\lambda }{p-\eta }z{h}^{\prime }\left(z\right)+\left(e^{i\lambda }-\varphi cos\lambda +\frac{\varphi cos\lambda }{p-\eta }\right)h\left(z\right)-\frac{\eta \left(\varphi cos\lambda \right)}{p-\eta }\hfill \\ & =B\left(z\right)z{h}^{\prime }\left(z\right)+C\left(z\right)h\left(z\right)+D\left(z\right),\hfill \end{array}$$

where:

$$B\left(z\right)=\frac{\varphi cos\lambda }{p-\eta },$$

$$C\left(z\right)=\frac{\left(\left(p-\eta \right)e^{i\lambda }-\left(p-\eta \right)\varphi cos\lambda +\varphi cos\lambda \right)}{p-\eta }$$

and:

$$D\left(z\right)=\frac{\eta \left(\varphi cos\lambda \right)}{p-\eta }.$$

Now, if we apply Lemma 1 with $A=0,$ we get:

$$\frac{f^{\prime }\left(z\right)}{{\psi }^{\prime }\left(z\right)}=h\left(z\right)\prec p_k\left(z\right).$$

(20)

Furthermore, from (18), we have:

$$p\left(z\right)+z{p}^{\prime }\left(z\right)q\left(z\right)\prec p_k\left(z\right).$$

By using Lemma 2 on (20), we obtain the desired result. This completes the proof of Theorem 5. □

4. Conclusions

Using the idea of spiral-like and close-to-convex functions, we have introduced Mocanu-type functions associated with conic domains. We have derived some interesting results such as sufficiency criteria, inclusion results, and integral-preserving properties. We have also proven that the our newly-defined function classes are closed under the famous Libera operator.