Bohr Radius Problems for Some Classes of Analytic Functions Using Quantum Calculus Approach

Abstract

The main purpose of this investigation is to use quantum calculus approach and obtain the Bohr radius for the class of q-starlike (q-convex) functions of order $\alpha$. The Bohr radius is also determined for a generalized class of q-Janowski starlike and q-Janowski convex functions with negative coefficients.

1. Introduction

Let $\mathbb{D}:=\{z:\in \mathbb{C}:|z|<1\}$ be the open unit disc in $\mathbb{C}$. Suppose $\mathcal{A}$ denote the class of analytic functions in $\mathbb{D}$ normalized by $f\left(0\right)=0=f^{\prime }\left(0\right)-1$. Also, let $\mathcal{S}$ be the subclass of $\mathcal{A}$ consisting of univalent functions in $\mathbb{D}$.

Suppose $\mathcal{H}(\mathbb{D},\Omega )$ is the class of analytic functions mapping open unit disc $\mathbb{D}$ into a domain $\Omega$. Harald Bohr [1] in 1914 proved that if a function f of the form $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ belong to $\mathcal{H}(\mathbb{D},\mathbb{D})$, then ${\sum }_{n=0}^{\infty }\left|a_n{z}^n\right|\le 1$ in the disc $\left|z\right|\le k,$ where $k\ge 1/6$. As reported by Bohr in [1], Riesz, Schur and Wiener discovered that $\left|z\right|\le k$ is actually true for $0\le k\le 1/3$ and that $1/3$ is the best possible. The number $1/3$ is commonly called the "Bohr radius" for the class of analytic self-maps f in $\mathbb{D},$ while the inequality ${\sum }_{n=0}^{\infty }\left|a_n{z}^n\right|\le 1$ is known as the "Bohr inequality". Later on, extensions of Bohr inequality and their proofs were given in [2,3,4]. Note that Bohr Radius is somewhat whimsical, for physicists consider the Bohr Radius $a_0$ of the hydrogen atom to be a fundamental constant, that is, $4\pi ϵh^2/m_e{e}^2$, or about $0.529$A. The physicists Bohr Radius is named for Niels Bohr, a founder of the Quantum Theory and 1922 recipient of the Nobel Prize for physics.

The Bohr inequality has emerged as an active area of research after Dixon [5] used it to disprove a conjecture in Banach algebra. Using the Euclidean distance, denoted by d, the Bohr inequality ${\sum }_{n=0}^{\infty }\left|a_n{z}^n\right|\le 1$ for a function f of the form $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ can be written as

$$\begin{array}{cc}\hfill {\displaystyle \sum _{n=0}^{\infty }}|a_n{z}^n|\le 1& \iff {\displaystyle \sum _{n=1}^{\infty }}|a_n{z}^n|\le 1-|a_0|\hfill \\ \hfill & \iff d\left({\displaystyle \sum _{n=0}^{\infty }}|a_n{z}^n|,|a_0|\right)={\displaystyle \sum _{n=1}^{\infty }}|a_n{z}^n|\le 1-|a_0|=1-|f\left(0\right)|\hfill \\ \hfill & \iff d\left({\displaystyle \sum _{n=0}^{\infty }}|a_n{z}^n|,|a_0|\right)\le d(f\left(0\right),\partial \mathbb{D}).\hfill \end{array}$$

where $\partial \mathbb{D}$ is the boundary of the disc $\mathbb{D}$. Thus, the concept of the Bohr inequality for a function $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$, defined in $\mathbb{D}$, can be generalized by

$$d\left(\sum _{n=0}^{\infty }|a_n{z}^n|,|f\left(0\right)|\right)=\sum _{n=1}^{\infty }|a_n{z}^n|\le d(f\left(0\right),\partial f\left(\mathbb{D}\right)).$$

(1)

Accordingly, the Bohr radius for a class $\mathcal{M}$ consisting of analytic functions f of the form $f\left(z\right)={\sum }_{n=0}^{\infty }{a}_n{z}^n$ in the disc $\mathbb{D}$ is the largest $r^{*}>0$ such that every function $f\in \mathcal{M}$ satisfies the inequality (1) for all $|z|=r\le r^{*}$. In this case, the class $\mathcal{M}$ is said to satisfy a Bohr phenomenon.

Quantum calculus (or q-calculus) is an approach or a methodology that is centered on the idea of obtaining q-analogues without the use of limits. This approach has a great interest due to its applications in various branches of mathematics and physics, such as, the areas of ordinary fractional calculus, optimal control problems, q-difference, q-integral equations and q-transform analysis. Jackson [6] intoduced the q-derivative (or q-difference, or Jackson derivative) denoted by $D_q$, $q\in (0,1)$, which is defined in a given subset of $\mathbb{C}$ by

$$\left(D_{q}f\right)\left(z\right)=\left\{\begin{array}{cc}\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z},\hfill & \mathrm{if}z\ne 0\hfill \\ f^{\prime }\left(0\right),\hfill & \mathrm{if}z=0\hfill \end{array}\right.$$

(2)

provided $f^{\prime }\left(0\right)$ exists. If f is a function defined in a subset of the complex plane $\mathbb{C}$, then (2) yields

$$\underset{q\to 1^{-}}{lim}\left(D_{q}f\right)\left(z\right)=\underset{q\to 1^{-}}{lim}\frac{f\left(z\right)-f\left(qz\right)}{(1-q)z}=f^{\prime }\left(z\right).$$

It is easy to see that if $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n$, then by using (2) we have

$$\left(D_{q}f\right)\left(z\right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{z}^{n-1},$$

$$D_q\left(z{D}_{q}f\left(z\right)\right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q^2{a}_n{z}^{n-1},$$

$$D_q^{2}f\left(z\right)=D_q\left(D_{q}f\left(z\right)\right)=\sum _{n=2}^{\infty }{\left[n\right]}_q^2{a}_n{z}^{n-2},$$

where ${\left[n\right]}_q$ is given by

$${\left[n\right]}_q=\frac{1-q^n}{1-q},q\in (0,1).$$

It is a routine to check that

$$D_q\left(z{D}_{q}f\left(z\right)\right)=D_{q}f\left(z\right)+z{D}_q^{2}f\left(z\right).$$

In 1869, Thomae introduced the particular q-integral [7] which is defined as

$$\underset{0}{\overset{1}{\int }}f\left(t\right)d_{q}t=(1-q)\sum _{n=0}^{\infty }{q}^{n}f\left(q^n\right),$$

provided the q-series converges. Later on, Jackson [8] defined the general q-integral as follows:

$$\underset{a}{\overset{b}{\int }}f\left(t\right)d_{q}t=\underset{0}{\overset{b}{\int }}f\left(t\right)d_{q}t-\underset{0}{\overset{a}{\int }}f\left(t\right)d_{q}t,$$

where

$$\underset{0}{\overset{a}{\int }}f\left(t\right)d_{q}t=a(1-q)\sum _{n=0}^{\infty }{q}^{n}f\left(a{q}^n\right),$$

provided the q-series converges. Also note that

$$D_q\underset{0}{\overset{x}{\int }}f\left(t\right)d_{q}t=f\left(x\right)\mathrm{and}\underset{0}{\overset{x}{\int }}{D}_{q}f\left(t\right)d_{q}t=f\left(x\right)-f\left(0\right),$$

where the second equality holds if f is continuous at $x=0$.

The q-calculus plays an important role in the investigation of several subclasses of $\mathcal{A}$. A firm footing of the q-calculus in the context of geometric function theory and its usages involving the basic (or q-) hypergeometric functions in geometric function theory was actually made in a book chapter by Srivastava (see, for details [9]; see also [10]). In 1990, Ismail et al. [11] introduced a connection between starlike (convex) functions and the q-calculus by introducing a q-analog of starlike (convex) functions. They generalized a well-known class of starlike functions, called the class of q-starlike functions denoted by ${\mathcal{S}}_q^{*}$, consisting of functions $f\in \mathcal{A}$ satisfying the inequality

$$\left|\frac{z\left(D_{q}f\right)\left(z\right)}{f\left(z\right)}-\frac{1}{1-q}\right|\le \frac{1}{1-q},z\in \mathbb{D}.$$

Baricz and Swaminathan [12] introduced a q-analog of convex functions, denoted by ${\mathcal{C}}_q$, satisfying the relation

$$f\in {\mathcal{C}}_q\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}z\left(D_{q}f\right)\in {\mathcal{S}}_q^{*}.$$

Recently Srivastava et al. [13] (see also [14]) successfully combined the concept of Janowski [15] and the above mentioned q-calculus and introduced the class ${\mathcal{S}}_q^{*}[A,B]$ and ${\mathcal{C}}_q[A,B]$, $-1\le B<A\le 1$, $q\in (0,1)$, given by

$${\mathcal{S}}_q^{*}[A,B]:=\left\{f\in \mathcal{A}:\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec \frac{(A+1)z+2+(A-1)qz}{(B+1)z+2+(B-1)qz}\right\},$$

and

$${\mathcal{C}}_q[A,B]:=\left\{f\in \mathcal{A}:1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\prec \frac{(A+1)z+2+(A-1)qz}{(B+1)z+2+(B-1)qz}\right\}$$

respectively, where ≺ denotes subordination. As $q\to 1^{-}$, ${\mathcal{S}}_q^{*}[A,B]$ and ${\mathcal{C}}_q[A,B]$ yield respectively the classes ${\mathcal{S}}^{*}[A,B]$ and $\mathcal{C}[A,B]$ defined by Janowski [15]. For various choices of A and B, these classes reduce to well-known subclasses of q-starlike and q-convex functions. For instance, with $0\le \alpha <1$, ${\mathcal{S}}_q^{*}\left(\alpha \right):={\mathcal{S}}_q^{*}[1-2\alpha ,-1]$ is the class of q-starlike functions of order $\alpha$, introduced by Agrawal and Sahoo [16]. Motivated by the authors in [16], Agrawal [17] defined a q-analog of convex functions of order $\alpha$, $0\le \alpha <1$, ${\mathcal{C}}_q\left(\alpha \right):={\mathcal{C}}_q[1-2\alpha ,-1]$, satisfying

$$f\in {\mathcal{C}}_q\left(\alpha \right)\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}z\left(D_{q}f\right)\in {\mathcal{S}}_q^{*}\left(\alpha \right).$$

(3)

Note that ${\mathcal{S}}_q^{*}[1,-1]\equiv {\mathcal{S}}_q^{*}$ and ${\mathcal{C}}_q[1,-1]\equiv {\mathcal{C}}_q$.

In recent years, there is a great development of geometric function theory because of using quantum calculus approach. In particular, Srivastava et al. [18] found distortion and radius of univalence and starlikenss for several subclasses of q-starlike functions with negative coefficients. They [19] also determined sufficient conditions and containment results for the different types of k-uniformly q-starlike functions. Naeem et al. [20] investigated subfamilies of q-convex functions and q-close to convex functions with respect to the Janowski functions connected with q-conic domain which explored some important geometric properties such as coefficient estimates, sufficiency criteria and convolution properties of these classes. For a survey on the use of quantum calculus approach in mathematical sciences and its role in geometric function theory, one may refer to [21]. In addition, one may refer to a survey-cum-expository article written by Srivastava [22] where he explored the mathematical application of q-calculus, fractional q- calculus and fractional q-differential operators in geometric function theory.

In this paper, we investigate Bohr radius problems for the classes ${\mathcal{S}}_q^{*}\left(\alpha \right)$ and ${\mathcal{C}}_q\left(\alpha \right)$, respectively, in Section 2 and Section 3. In Section 4, we define and investigate the Bohr radius problem for a generalized class, ${\mathcal{TP}}_q(\lambda ,A,B)$, of functions with negative coefficients, where $q\in (0,1)$, $\lambda \in [0,1]$ and $-1\le B<A\le 1$. In particular, we also define and obtain sharp Bohr radius for the class of the q-Janowski functions with negative coefficients in Section 4.

2. The Bohr Radius for the Class ${\mathit{S}}_{\mathit{q}}^{*}\left(\mathit{\alpha }\right)$

To find the Bohr radius for the class $S_q^{*}\left(\alpha \right)$, we first need the following four lemmas.

Lemma 1

([23] (Theorem 2.5, p. 1511)). For $q\in (0,1)$, suppose $a,b,c$ are non-negative real numbers satisfying $0\le 1-aq\le 1-cq$ and $0<1-b\le 1-c$. Then there exists a non-decreasing function $\mu :[0,1]\to [0,1]$ with $\mu \left(1\right)-\mu \left(0\right)=1$ such that

$$\frac{w\varphi (q,q,q^2,q,w)}{\varphi (q^0,q,q^2,q,w)}={\int }_0^1\frac{w}{1-tw}d\mu \left(t\right),$$

where $\varphi (a,b;c;q,z)$ is a hypergeometric function (see [24,25]) given by

$$\varphi (a,b;c;q,z)=\sum _{n=0}^{\infty }\frac{{(a;q)}_n{(b;q)}_n}{{(c;q)}_n{(q;q)}_n}{z}^n$$

and ${(a;q)}_0=1,{(a;q)}_n=(1-a)(1-aq)(1-a{q}^2)\cdots (1-a{q}^{n-1})$, which is analytic in the cut-plane $\mathbb{C}\backslash [1,\infty ]$ and maps both the unit disc and the half-plane $\{z\in \mathbb{C}:Rez<1\}$ univalently onto domains convex in the direction of the imaginary axis.

Lemma 2

([16] (Theorem 1.1, p. 17)). If $f\in \mathcal{A}$, then $f\in {\mathcal{S}}_q^{*}\left(\alpha \right)$ if and only if there exists a probability measure μ supported on the circle such that

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}=1+{\int }_{|\sigma |=1}\sigma z{F}_{q,\alpha }^{\prime }\left(\sigma z\right)d\mu \left(\sigma \right),$$

where

$$F_{q,\alpha }\left(z\right)=\sum _{n=1}^{\infty }\frac{-2}{1-q^n}ln\left(\frac{q}{1-\alpha (1-q)}\right)z^n,z\in \mathbb{D}.$$

Lemma 3

(Distortion theorem). Let $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n=zh\left(z\right)\in {\mathcal{S}}_q^{*}\left(\alpha \right)$. Then

$$exp\left(F_{q,\alpha }(-r)\right)\le |h\left(z\right)|\le exp\left(F_{q,\alpha }\left(r\right)\right).$$

Proof. 

Let $f\in {\mathcal{S}}_q^{*}\left(\alpha \right)$. By Lemma 2, there exists a probability measure $\mu$ supported on the unit circle such that

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}=1+{\int }_{|\sigma |=1}\sigma z{F}_{q,\alpha }^{\prime }\left(\sigma z\right)d\mu \left(\sigma \right),$$

where

$$F_{q,\alpha }\left(z\right)=\sum _{n=1}^{\infty }\frac{-2ln\left(\frac{q}{1-\alpha (1-q)}\right)}{1-q^n}{z}^n,z\in \mathbb{D}.$$

Integrating and then taking exponential on both sides, we have

$$f\left(z\right)=zexp\left({\int }_{|\sigma |=1}{F}_{q,\alpha }\left(\sigma z\right)d\mu \left(\sigma \right)\right).$$

Since $f\left(z\right)=zh\left(z\right)\in S_q^{*}\left(\alpha \right)$, it follows that

$$|h\left(z\right)|=exp\left(Re{\int }_{|\sigma |=1}{F}_{q,\alpha }\left(\sigma z\right)d\mu \left(\sigma \right)\right).$$

Thus

$$\begin{array}{cc}\hfill ln|h(z)|& =Re{\int }_{|\sigma |=1}{F}_{q,\alpha }\left(\sigma z\right)d\mu \left(\sigma \right)\hfill \\ \hfill & =-2ln\left(\frac{q}{1-\alpha (1-q)}\right)Re{\int }_{|\sigma |=1}{\displaystyle \sum _{n=1}^{\infty }}\frac{{\left(\sigma z\right)}^n}{1-q^n}d\mu \left(\sigma \right)\hfill \\ \hfill & =\frac{-2}{1-q}ln\left(\frac{q}{1-\alpha (1-q)}\right)Re{\int }_{|\sigma |=1}\left(\sigma z\varphi (q,q,q^2,q,\sigma z)\right)d\mu \left(\sigma \right)\hfill \\ \hfill & =\frac{-2}{1-q}ln\left(\frac{q}{1-\alpha (1-q)}\right)Re{\int }_0^{2\pi }\left(\left(e^{i\theta }z\right)\varphi (q,q,q^2,q,e^{i\theta }z)\right)d\mu \left(\theta \right)\hfill \\ \hfill & =\frac{-2}{1-q}ln\left(\frac{q}{1-\alpha (1-q)}\right)Re{\int }_0^{2\pi }\left(w\varphi (q,q,q^2,q,w)\right)d\mu \left(\theta \right),w=e^{i\theta }z\in \mathbb{D}\hfill \\ \hfill & =\frac{-2}{1-q}ln\left(\frac{q}{1-\alpha (1-q)}\right)Re{\int }_0^{2\pi }\frac{w\varphi (q,q,q^2,q,w)}{\varphi (q^0,q,q^2,q,w)}d\mu \left(\theta \right),\hfill \end{array}$$

(4)

where $\varphi (a,b;c;q,z)$ is the hypergeometric function defined in Lemma 1. By Lemma 1, we have

$$\frac{w\varphi (q,q,q^2,q,w)}{\varphi (q^0,q,q^2,q,w)}={\int }_0^1\frac{w}{1-tw}d\mu \left(t\right).$$

(5)

Let

$$\begin{array}{cc}\hfill g\left(r{e}^{i\psi }\right)& =Re\frac{w}{1-tw},w=r{e}^{i\psi }\hfill \\ \hfill & =Re\frac{r(cos\psi +isin\psi )}{1-tr(cos\psi +isin\psi )}\hfill \\ \hfill & =\frac{rcos\psi (1-trcos\psi )-t{r}^2{sin}^2\psi }{1+r^2{t}^2-2trcos\psi }.\hfill \end{array}$$

A routine calculation shows that

$$\underset{\psi }{min}g\left(r{e}^{i\psi }\right)=g(-r)\mathrm{and}\underset{\psi }{max}g\left(r{e}^{i\psi }\right)=g\left(r\right).$$

Thus

$$\underset{|w|\le r}{min}Re\frac{w}{1-tw}=\frac{-r}{1+rt}\mathrm{and}\underset{|w|\le r}{max}Re\frac{w}{1-tw}=\frac{r}{1-rt}.$$

(6)

By (4)–(6), it follows that

$$\begin{array}{cc}\hfill ln|h(z)|& \ge \frac{-2}{1-q}ln\left(\frac{q}{1-\alpha (1-q)}\right){\int }_{|\sigma |=1}(-r\varphi (q,q,q^2,q,-r))d\mu \left(\sigma \right)\hfill \\ \hfill & \ge \frac{-2}{1-q}ln\left(\frac{q}{1-\alpha (1-q)}\right)(-r\varphi (q,q,q^2,q,-r))\hfill \\ \hfill & =F_{q,\alpha }(-r)\hfill \end{array}$$

(7)

and

$$\begin{array}{cc}\hfill ln|h(z)|& \le {\int }_{|\sigma |=1}{F}_{q,\alpha }\left(r\right)d\mu \left(\sigma \right)\hfill \\ \hfill & =F_{q,\alpha }\left(r\right).\hfill \end{array}$$

(8)

By (7) and (8), we have $exp\left(F_{q,\alpha }(-r)\right)\le |h\left(z\right)|\le exp\left(F_{q,\alpha }\left(r\right)\right).$ □

Remark 1.

As $q\to 1^{-}$, Lemma 3 yields the corresponding distortion theorem [26] (Theorem 8, p. 117) for the class $S^{*}\left(\alpha \right)$.

Lemma 4

([16] (Theorem 1.3, p. 8)). Let

$$G_{q,\alpha }\left(z\right)=zexp\left(F_{q,\alpha }\left(z\right)\right)=z+\sum _{n=2}^{\infty }{c}_n{z}^n.$$

Then $G_{q,\alpha }\left(z\right)\in {\mathcal{S}}_q^{*}\left(\alpha \right)$. However, if $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n\in {\mathcal{S}}_q^{*}\left(\alpha \right)$, then $|a_n|\le c_n$ with equality holding for all n if and only if f is a rotation of $G_{q,\alpha }$.

Theorem 1.

Let $\varphi \left(z\right)={\sum }_{n=1}^{\infty }{\varphi }_n{z}^n$ and $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n=zexp\left(\varphi \left(z\right)\right)\in {\mathcal{S}}_q^{*}\left(\alpha \right)$. Then

$$|z|+\sum _{n=2}^{\infty }|a_n{||z|}^n\le d(0,\partial f\left(\mathbb{D}\right))$$

for $|z|\le r^{*}$, where $r^{*}\in (0,1)$ is the unique root of the equation

$$rexp\left(F_{q,\alpha }\left(r\right)\right)=exp\left(F_{q,\alpha }(-1)\right).$$

The radius is sharp.

Proof. 

Let $f\in {\mathcal{S}}_q^{*}\left(\alpha \right)$. Proceeding as in proof of [16] (Theorem 1.3, p. 8), it is easy to see that coefficients bound for the function $\varphi \left(z\right)={\sum }_{n=1}^{\infty }{\varphi }_n{z}^n$ are given by

$$|{\varphi }_n|\le \frac{-2ln\left(\frac{q}{1-\alpha (1-q)}\right)}{1-q^n}.$$

(9)

For $|z|=r\le r^{*}$, using Lemma 3 and inequality (9), it follows that

$$\begin{array}{cc}\hfill d(0,\partial f\left(\mathbb{D}\right))={\displaystyle \underset{|z|\to 1^{-}}{lim}}inf|f\left(z\right)-f\left(0\right)|& ={\displaystyle \underset{|z|\to 1^{-}}{lim}}inf\frac{|f(z)|}{|z|}\ge exp{F}_{q,\alpha }(-1)\hfill \\ \hfill & \ge rexp{F}_{q,\alpha }\left(r\right)\hfill \\ \hfill & =rexp\left({\displaystyle \sum _{n=1}^{\infty }}\frac{-2ln\left(\frac{q}{1-\alpha (1-q)}\right)}{1-q^n}{r}^n\right)\hfill \\ \hfill & \ge |z|+{\displaystyle \sum _{n=2}^{\infty }}|a_n{||z|}^n\hfill \end{array}$$

if and only if

$$rexp\left(F_{q,\alpha }\left(r\right)\right)\le exp{F}_{q,\alpha }(-1).$$

In order to prove that the radius is sharp, let

$$G_{q,\alpha }\left(z\right):=zexp\left(F_{q,\alpha }\left(z\right)\right),$$

where

$$F_{q,\alpha }\left(z\right)=\sum _{n=1}^{\infty }\frac{-2}{1-q^n}ln\left(\frac{q}{1-\alpha (1-q)}\right)z^n,z\in \mathbb{D}.$$

By Lemma 4, it follows that $G_{q,\alpha }\in {\mathcal{S}}_q^{*}\left(\alpha \right)$. For $|z|=r^{*}$, we obtain

$$\begin{array}{cc}\hfill |z|+{\displaystyle \sum _{n=2}^{\infty }}|a_n{||z|}^n& =r^{*}exp\left({\displaystyle \sum _{n=1}^{\infty }}\frac{-2}{1-q^n}ln\left(\frac{q}{1-\alpha (1-q)}\right){\left(r^{*}\right)}^n\right)\hfill \\ \hfill & =r^{*}exp{F}_{q,\alpha }\left(r^{*}\right)\hfill \\ \hfill & =exp{F}_{q,\alpha }(-1)\hfill \\ \hfill & ={\displaystyle \underset{|z|\to 1^{-}}{lim}}inf\frac{|G_{q,\alpha }\left(z\right)|}{|z|}\hfill \\ \hfill & ={\displaystyle \underset{|z|\to 1^{-}}{lim}}inf|G_{q,\alpha }\left(z\right)-f\left(0\right)|\hfill \\ \hfill & =d(0,G_{q,\alpha }\left(\mathbb{D}\right)).\hfill \end{array}$$

 □

Remark 2.

For $\alpha =0$, Theorem 1 yields the corresponding results found in [27] for the class $S_q^{*}$.

Remark 3.

Theorem 1 with letting $q\to 1^{-}$ leads to the Bohr radius for the class of starlike functions of order α, $0\le \alpha <1$. Bhowmik and Das [28] (Theorem 3, p. 1093) found the Bohr radius for $S^{*}\left(\alpha \right)$ with $\alpha \in [0,1/2]$.

3. The Bohr Radius for the Class ${\mathit{C}}_{\mathit{q}}\left(\mathit{\alpha }\right)$

In the present section, we obtain the sharp Bohr radius for the class of q-convex functions of order $\alpha$, $0\le \alpha <1$.

Lemma 5

([17] (Theorem 2.9, p. 5)). Let

$$E_q\left(z\right):=\underset{0}{\overset{z}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t=z+\sum _{n=2}^{\infty }\left(\frac{1-q}{1-q^n}{c}_n{z}^n\right),$$

where $c_n$ is the nth coefficient of the function $zexp\left(F_{q,\alpha }\left(z\right)\right)$. Then $E_q\in {\mathcal{C}}_q\left(\alpha \right)$ for $0\le \alpha <1$. Moreover, if $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n\in {\mathcal{C}}_q\left(\alpha \right)$, then $|a_n|\le ((1-q)/(1-q^n))c_n$, with equality holding for all n if and only if f is a rotation of $E_q$.

Theorem 2.

The Bohr radius for the class ${\mathcal{C}}_q\left(\alpha \right)$ is $r^{*}$, where $r^{*}\in (0,1]$ is the unique root of the equation

$$\underset{0}{\overset{r}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t=\underset{0}{\overset{1}{\int }}exp\left(F_{q,\alpha }(-t)\right)d_{q}t.$$

The radius is sharp.

Proof. 

Let $f\in {\mathcal{C}}_q\left(\alpha \right)$. Then, by (3), $z\left(D_{q}f\right)\left(z\right)\in {\mathcal{S}}_q^{*}\left(\alpha \right).$ It follows from Lemma 3 that

$$exp\left(F_{q,\alpha }(-r)\right)\le |\left(D_{q}f\right)\left(z\right)|\le exp\left(F_{q,\alpha }\left(r\right)\right).$$

Taking q-integral of all the inequalities, we have

$$\underset{0}{\overset{r}{\int }}exp\left(F_{q,\alpha }(-t)\right)d_{q}t\le |f\left(z\right)|\le \underset{0}{\overset{r}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t.$$

(10)

Since $f\left(z\right)=z+{\sum }_{n=2}^{\infty }{a}_n{z}^n\in {\mathcal{C}}_q\left(\alpha \right)$, Lemma 5 yields the coefficients bound for the function f given by

$$|a_n|\le \frac{1-q}{1-q^n}{c}_n,$$

(11)

where inequality holds for all n if and only if f is a rotation of

$$E_q\left(z\right)=\underset{0}{\overset{z}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t=z+\sum _{n=2}^{\infty }\left(\frac{1-q}{1-q^n}\right)c_n{z}^n$$

and where $c_n$ is the $nth$ coefficient of $zexp\left(F_{q,\alpha }\left(z\right)\right)$.

By (10) and (11), we have

$$\begin{array}{cc}\hfill r+{\displaystyle \sum _{n=2}^{\infty }}|a_n|r^n& \le r+{\displaystyle \sum _{n=2}^{\infty }}\frac{1-q}{1-q^n}{c}_n{r}^n\hfill \\ \hfill & =\underset{0}{\overset{r}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t\le \underset{0}{\overset{1}{\int }}exp\left(F_{q,\alpha }(-t)\right)d_{q}t\le d(0,\partial f\left(\mathbb{D}\right))\hfill \end{array}$$

if and only if

$$\underset{0}{\overset{r}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t\le \underset{0}{\overset{1}{\int }}exp\left(F_{q,\alpha }(-t)\right)d_{q}t.$$

Now, consider the function

$$E_q\left(z\right):=\underset{0}{\overset{z}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t=z+\sum _{n=2}^{\infty }\left(\frac{1-q}{1-q^n}\right)c_n{z}^n.$$

It follows from Lemma 5 that the function $E_q\left(z\right)\in {\mathcal{C}}_q\left(\alpha \right)$. At $|z|=r^{*}$, we have

$$\begin{array}{cc}\hfill r^{*}+{\displaystyle \sum _{n=2}^{\infty }}|a_n|{\left(r^{*}\right)}^n& =r^{*}+{\displaystyle \sum _{n=2}^{\infty }}\frac{1-q}{1-q^n}{c}_n{\left(r^{*}\right)}^n\hfill \\ \hfill & =\underset{0}{\overset{r^{*}}{\int }}exp\left(F_{q,\alpha }\left(t\right)\right)d_{q}t=\underset{0}{\overset{1}{\int }}exp\left(F_{q,\alpha }(-t)\right)d_{q}t=d(0,\partial E_q\left(\mathbb{D}\right))\hfill \end{array}$$

which shows that the Bohr radius $r^{*}$ is sharp for the class ${\mathcal{C}}_q\left(\alpha \right)$. □

Putting $\alpha =0$ in Theorem 2, we obtain the Bohr radius for the class ${\mathcal{C}}_q$ of q-convex functions.

Corollary 1

([27] (Theorem 2, p. 111)). The Bohr radius for the class ${\mathcal{C}}_q$ is $r^{*}$, where $r^{*}\in (0,1]$ is the unique root of

$$\underset{0}{\overset{r}{\int }}exp\left(F_{q,0}\left(t\right)\right)d_{q}t=\underset{0}{\overset{1}{\int }}exp\left(F_{q,0}(-t)\right)d_{q}t.$$

The radius is sharp.

If $q\to 1^{-}$, then Corollary 1 yields the Bohr radius for the class $\mathcal{C}$ of convex functions, that is, $r^{*}=1/3$. The same Bohr radius for general convex functions had been earlier obtained by Aizenberg in [29] (Thoerem 2.1).

4. The Bohr Radius Problems for the Class ${\mathcal{TP}}_{\mathit{q}}(\mathit{\lambda },\mathit{A},\mathit{B})$

In 1975, Silverman [30] investigated two new subclasses of the family $\mathcal{T}$, where

$$\mathcal{T}=\{f\in \mathcal{S}:f\left(z\right)=z-\sum _{n=2}^{\infty }|a_n|z^n,z\in \mathbb{D}\}.$$

Recently, Altıntaş and Mustafa [31] introduced a generalized class, ${\mathcal{TP}}_q(\lambda ,A,B),q\in (0,1),\lambda \in [0,1],-1\le B<A\le 1$, given by

$${\mathcal{TP}}_q(\lambda ,A,B)=\left\{f\in \mathcal{T}:\frac{z{D}_{q}f\left(z\right)+\lambda z^2{D}_q^{2}f\left(z\right)}{\lambda z{D}_{q}f\left(z\right)+(1-\lambda )f\left(z\right)}\prec \frac{1+Az}{1+Bz},z\in \mathbb{D}\right\}.$$

For $\lambda =0$, this class reduces to the class ${\mathcal{TS}}_q^{*}[A,B]$ of q-Janowski starlike functions with negative coefficients defined by

$${\mathcal{TS}}_q^{*}[A,B]=\left\{f\in \mathcal{T}:\frac{z{D}_{q}f\left(z\right)}{f\left(z\right)}\prec \frac{1+Az}{1+Bz},z\in \mathbb{D}\right\}.$$

On the other hand, the case $\lambda =1$ yields the class ${\mathcal{TC}}_q[A,B]$ of q-Janowski convex functions, defined by

$${\mathcal{TC}}_q[A,B]=\left\{f\in \mathcal{T}:1+\frac{z{D}_q^{2}f\left(z\right)}{D_{q}f\left(z\right)}\prec \frac{1+Az}{1+Bz},z\in \mathbb{D}\right\}.$$

As $q\to 1^{-}$, ${\mathcal{TS}}_q^{*}[A,B]$ and ${\mathcal{TC}}_q[A,B]$ reduce respectively to ${\mathcal{TS}}^{*}[A,B]$ and $\mathcal{TC}[A,B]$ studied initially in [32]. Note that the classes ${\mathcal{TS}}^{*}\left(\alpha \right)\equiv \underset{q\to 1^{-}}{lim}{\mathcal{TS}}_q^{*}[1-2\alpha ,-1]$ and $\mathcal{TC}\left(\alpha \right)\equiv \underset{q\to 1^{-}}{lim}{\mathcal{TC}}_q[1-2\alpha ,-1]$ were defined and studied by Silverman [30] in 1975.

In the present section, we will first investigate the sharp Bohr radius for the class ${\mathcal{TP}}_q(\lambda ,A,B)$, $q\in (0,1),\lambda \in [0,1]$ which in particular gives the Bohr radius for the classes ${\mathcal{TS}}_q^{*}[A,B]$ and ${\mathcal{TC}}_q[A,B]$. However, in order to obtain Bohr radius, we first need some results given here in two lemmas.

Note that there is a typing error in the statement of [31] (Theorem 3.1, p. 993) (replace $\alpha$ by $\beta$). The correct statement in Lemma 6 is as follows:

Lemma 6

([31] (Theorem 3.1, p. 993)). If $f\in {\mathcal{TP}}_q(\lambda ,A,B)$, $q\in (0,1)$, $\lambda \in [0,1]$, then

$$r-\frac{1-\beta }{({[2]}_q-\beta )[1+({[2]}_q-1)\lambda ]}{r}^2\le |f(z)|\le r+\frac{1-\beta }{({[2]}_q-\beta )[1+({[2]}_q-1)\lambda ]}{r}^2$$

where $\beta =(1-A)/(1-B),-1\le B<A\le 1$, with equality for the function

$$f(z)=z-\frac{1-\beta }{({[2]}_q-\beta )[1+({[2]}_q-1)\lambda ]}{z}^2,|z|=r.$$

Lemma 7

([31] (Theorem 2.8, p. 991)). If $f\in {\mathcal{TP}}_q(\lambda ,A,B)$, $q\in (0,1)$, $\lambda \in [0,1]$, then the following conditions are satisfied:

$$\sum _{n=2}^{\infty }|a_n|\le \frac{1-\beta }{({[n]}_q-\beta )(1+({[n]}_q-1)\lambda )}$$

$$\sum _{n=2}^{\infty }{[n]}_q|a_n|\le \frac{(1-\beta ){[n]}_q}{({[n]}_q-\beta )(1+({[n]}_q-1)\lambda )},n=2,3,\cdots ,$$

where $\beta =(1-A)/(1-B),-1\le B<A\le 1$. The results obtained here are sharp.

Theorem 3.

If $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in {\mathcal{TP}}_q(\lambda ,A,B)$ where $q\in (0,1)$, $\lambda \in [0,1]$, $\beta =(1-A)/(1-B)$ and $c=q(\lambda +1+q\lambda -\beta \lambda )$, then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$r^{*}=\frac{2c}{1-\beta +c+\sqrt{4(1-\beta )c+{(1-\beta +c)}^2}}.$$

The radius $r^{*}$ is the sharp Bohr radius for class ${\mathcal{TP}}_q(\lambda ,A,B)$.

Proof. 

It follows from Lemma 6 that the distance between the origin and the boundary of $f(\mathbb{D})$ satisfies the inequality

$$d(0,\partial f(\mathbb{D}))\ge 1-\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}.$$

(12)

The given $r^{*}$ is the root of the equation

$$r^{*}+\frac{(1-\beta ){(r^{*})}^2}{(1+q-\beta )(1+q\lambda )}=1-\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}.$$

For $0<r\le r^{*}$, we have

$$r+\frac{(1-\beta )r^2}{(1+q-\beta )(1+q\lambda )}\le r^{*}+\frac{(1-\beta ){(r^{*})}^2}{(1+q-\beta )(1+q\lambda )}=1-\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}.$$

Using Lemma 7, it is easy to show that

$$\sum _{n=2}^{\infty }|a_n|\le \frac{1-\beta }{(1+q-\beta )(1+q\lambda )}.$$

The above inequality together with inequality (12) yield

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le r+\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}{r}^2\le 1-\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}\le d(0,\partial f(\mathbb{D})).$$

For sharpness, consider the function $f:\mathbb{D}\to \mathbb{C}$ defined by

$$f(z)=z-\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}{z}^2.$$

This function clearly belongs to ${\mathcal{TP}}_q(\lambda ,A,B)$. For $|z|=r^{*}$, we find

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|=r^{*}+\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}{(r^{*})}^2=1-\frac{1-\beta }{(1+q-\beta )(1+q\lambda )}=d(0,\partial f(\mathbb{D})).$$

 □

Putting $\lambda =0$ in Theorem 3, we get the sharp Bohr radius for the class ${\mathcal{TS}}_q^{*}[A,B]$.

Theorem 4.

If $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in {\mathcal{TS}}_q^{*}[A,B]$, $\beta =(1-A)/(1-B)$ and $-1\le B<A\le 1$, then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$r^{*}=\frac{2q}{1+q-\beta +\sqrt{1+6q+q^2-2\beta -6q\beta +{\beta }^2}}.$$

The radius $r^{*}$ is sharp.

Letting $A=1-2\alpha$ and $B=-1$ in Theorem 4, we obtain the sharp Bohr radius for the class of q-starlike functions of order $\alpha$, $0\le \alpha <1$, with negative coefficients.

Corollary 2.

Let $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in {\mathcal{TS}}_q^{*}(\alpha )$. Then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$r^{*}=\frac{2q}{1+q-\alpha +\sqrt{q^2+6q(1-\alpha )+{(1-\alpha )}^2}}.$$

When $q\to 1^{-}$ in Corollary 2, we obtain the following sharp Bohr radius for the class of starlike functions of order $\alpha$, $0\le \alpha <1$, with negative coefficients obtained by Ali et al. [33].

Corollary 3

([33] (Theorem 2.3)). If $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in {\mathcal{TS}}^{*}(\alpha )$, then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$r^{*}=\frac{2}{2-\alpha +\sqrt{8-8\alpha +{\alpha }^2}}.$$

The radius $r^{*}$ is the Bohr radius for ${\mathcal{TS}}^{*}(\alpha )$.

When $A=1$ and $B=-1$, Theorem 4 gives the following sharp Bohr radius for the class of q-starlike functions with negative coefficients.

Corollary 4.

If $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in {\mathcal{TS}}_q^{*}$, then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$r^{*}=\frac{2q}{1+q+\sqrt{1+6q+q^2}}.$$

When $A=1$, $B=-1$ and $q\to 1^{-}$, Theorem 4 gives the following sharp Bohr radius for the class of starlike functions with negative coefficients obtained by Ali et al. [33].

Corollary 5

([33]). The sharp Bohr radius for the class ${\mathcal{TS}}^{*}$ is $\sqrt{2}-1\simeq 0.414214$.

When $\lambda =1$, Theorem 3 gives the following sharp Bohr radius for the class of ${\mathcal{TC}}_q[A,B]$.

Theorem 5.

If $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in {\mathcal{TC}}_q[A,B]$, $\beta =(1-A)/(1-B)$ and $-1\le B<A\le 1$, then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$r^{*}=\frac{2q(2+q-\beta )}{1+2q+q^2-\beta -q\beta +\sqrt{4(1-\beta )\left(2q+q^2-q\beta \right)+{\left(q\beta -1-2q-q^2+\beta \right)}^2}}.$$

The result is sharp for the function

$$f(z)=z-\frac{1-\beta }{(1+q-\beta )(1+q)}{z}^2.$$

When $A=1-2\alpha$ and $B=-1$, Theorem 5 gives the sharp Bohr radius for the class of q-convex functions with negative coefficients.

Corollary 6.

The sharp Bohr radius for the class ${\mathcal{TC}}_q(\alpha )$ is

$$\frac{2q(2+q-\alpha )}{1+2q+q^2-\alpha -q\alpha +\sqrt{{(1+q)}^2{(1+q-\alpha )}^2+4q(2+q-\alpha )(1-\alpha )}}.$$

Letting $q\to 1^{-}$ in Corollary 6, we get the following sharp Bohr radius for the class of convex functions of order $\alpha$, $0\le \alpha <1$, with negative coefficients obtained by Ali et al. [33].

Corollary 7

([33] (Theorem 2.4)). If $f(z)=z-{\sum }_{n=2}^{\infty }|a_n|z^n\in \mathcal{TC}(\alpha )$, then

$$|z|+\sum _{n=2}^{\infty }|a_n{z}^n|\le d(0,\partial f(\mathbb{D}))$$

for $|z|<r^{*}$, where

$$\frac{3-\alpha }{2-\alpha +\sqrt{7-8\alpha +2{\alpha }^2}}.$$

The radius $r^{*}$ is the Bohr radius for $\mathcal{TC}(\alpha )$.

For $A=1$ and $B=-1$, Theorem 5 yields the sharp Bohr radius for the class of q-convex functions with negative coefficients.

Corollary 8.

The sharp Bohr radius for the class ${\mathcal{TC}}_q$ is

$$\frac{2q(2+q)}{1+2q+q^2+\sqrt{1+12q+10q^2+4q^3+q^4}}.$$

Letting $q\to 1^{-}$, $A=1$ and $B=-1$, Theorem 5 gives the sharp Bohr radius for the class of convex functions with negative coefficients by Ali et al. [33].

Corollary 9

([33]). The sharp Bohr radius for the class $\mathcal{TC}$ is $\sqrt{7}-2\simeq 0.645751$.