Relations of a planar domains bounded by hyperbolas with families of holomorphic functions

We consider a family of analytic and normalized functions with the property that \(zf'(z)/f(z)\) (or \(1+zf''(z)/f'(z)\)) lies in a domain bounded by a right branch of a hyperbola \(\rho =\rho (s)= ( 2 \cos \frac{\varphi }{s} )^{-s}\), where \(0< s\le 1\) and \(|\varphi |<(\pi s)/2\). A comprehensive characteristic of that families and relations with the well known families of univalent functions are presented. Some relevant examples are indicated.

Let \(\mathcal{H}\) denote the class of functions f holomorphic in the open unit disc \(\mathbb{D}= \{z\in \mathbb{C}: |z|<1 \}\) of the complex plane \(\mathbb{C}\), and with the power series

By \(\mathcal{S}\) we denote the subclass of \(\mathcal{H}\) consisting of univalent functions, and by \(\mathcal{ST}\), \(\mathcal{CV}\) the best known subclasses of \(\mathcal{S}\) containing functions starlike with respect to the origin and convex, respectively. Their geometric properties and various applications have caused rapid development of research in these directions. Many its subclasses have been introduced and investigated. The main feature of the new definitions is usually fact that either \(zf'(z)/f(z)\) or \(1+ zf''(z)/f'(z)\) is located in a some region contained in a right halfplane. A uniform approach to such definitions was proposed by Ma and Minda [7]. They introduced a special class of functions φ that map the unit disk onto the domains \(\varphi (\mathbb{D})\) symmetric with respect to the real axis, starlike with respect to \(\varphi (0)=1\), and with \(\varphi '(0)>0\). A family \(\mathcal{P}(\varphi )\) that consists of holomorphic functions p such that \(p(0)=1\) and \(p(\mathbb{D})\subset \varphi (\mathbb{D})\) (or in terms of subordination as \(p\prec \varphi \)) have also been defined in [7] (in connection with the function φ). The Carathéodory class \(\mathcal{P}=\{p(z)=1+p_{1}z+p_{2} z^{2} + \cdots , \Re p(z)>0, z\in \mathbb{D}\}\) is simply the class \(\mathcal{P}((1+z)/(1-z))\). The class of starlike univalent functions \(\mathcal{ST}\) (and convex functions \(\mathcal{CV}\), resp.) can be described as \(zf'/f\in \mathcal{P}((1+z)/(1-z))\) (or \(1+zf''/f'\in \mathcal{P}((1+z)/(1-z))\), resp.). It is seen that if φ varies, \(\mathcal{P}(\varphi )\) generates a number of well known subclasses of the Carathéodory class. In a such way, various similar subclasses of \(\mathcal{P}\) were defined and studied through a years. Leading examples of \(\mathcal{P} \) are \(\mathcal{P}(\alpha )=\mathcal{P}(\varphi _{\alpha }(z))=\mathcal{P}((1+(1-2\alpha )z)/(1-z)) \), \(\mathcal{P}_{\gamma }=\mathcal{P}(\varphi ^{\gamma }(z))=\mathcal{P}( ((1+z)/(1-z) )^{\gamma })\), \(\mathcal{P}(A,B)= \mathcal{P}(\varphi _{A,B}(z))=\mathcal{P}((1+Az)/(1+Bz))\), etc. Related families of starlike and convex functions of order α (\(0\le \alpha <1\)) (strongly starlike and convex of order γ (\(0<\gamma \le 1\)), resp.) were defined as follows:

and also the corresponding subclasses \(\mathcal{CV}(\alpha )\), \(\mathcal{CV}_{\gamma }\) of convex functions, for which \(1+zf''/f' \prec \varphi _{\alpha }\) or \(1+zf''/f'\prec \varphi ^{\gamma }(z)\), respectively (see, for example [1, 9, 11]). We note that \(\mathcal{ST}=\mathcal{ST}(0)=\mathcal{ST}_{1}\) and \(\mathcal{CV}= \mathcal{CV}(0)=\mathcal{CV}_{1}\).

An interesting families of the domains that are bounded by a conic sections were introduced and studied in [5, 6]. Such domains are related to k-starlike functions \(k\mbox{-} \mathcal{ST}\), and k-uniformly convex function, denoted \(k\mbox{-}\mathcal{UCV}\). A univalent function \(f\in k\mbox{-} \mathcal{ST}\) if the expression \(zf'/f\) satisfies

and \(f\in k\mbox{-}\mathcal{UCV}\) if \(\Re (1+zf''(z)/f'(z)) > k|zf''(z)/f'(z)| \) for \(z\in \mathbb{D}\). Equivalently, these families may be geometrically described by the fact that the expression \(p(z)=zf'(z)/f(z)\) (or \(p(z)=1+zf''(z)/f'(z)\), resp.) lies in a domain \(\varOmega _{k}\) for \(z\in \mathbb{D}\), where

We note that \(\varOmega _{k}\) is bounded by the conic sections of a shape depending on a parameter k and contained in a right halfplane. \(\partial \varOmega _{k}\) reduces to the imaginary axis for \(k=0\), a hyperbola for \(0< k<1\), parabola for \(k=1\), and finally to an ellipse, when \(1< k<\infty \). It was observed in [5, 6] that each \(\varOmega _{k}\) is symmetric with respect to the real axis and starlike with respect to the point 1. Additionally the functions \(p_{k}\), constructed in [6], see also [4], that map the unit disk onto \(\varOmega _{k}\) satisfy \(p_{k}(0)=1\), \(p_{k}'(0)>0\), for \(0\le k< \infty \), therefore fulfill the Ma and Minda assertions.

We also recall that the hyperbola considered in [5, 6] is given by

and corresponds to the parameter \(0< k<1\). Moreover, the points of intersection of \(\partial \varOmega _{k}\) with the real axis is \((u,0)=(k/(k+1),0)\) and the slope angle of asymptotes of \(\partial \varOmega _{k}\) to the real axis equals \(\arctan (1/k)\).

This work is intended as an attempt at defining a new subfamily of \(\mathcal{P}\) related to the domains bounded by a right branch of a hyperbolas \(H(s)\) for \(0< s\le 1\), where

It is easy to observe that the intersection point of \(H(s)\) and real axis is \((u,0)=(1/2^{s},0)\) and the slope angle to the real axis is equal \((\pi s)/2\). Taking into account these facts, we want to emphasize then the fact that none hyperbola \(\partial \varOmega _{k}\) reduces to \(H(s)\) and conversely. Indeed, considering a location of common points of the hyperbola \(H(s)\), hyperbola \(\partial \varOmega _{k}\) with the real axis and, we obtain the relation \(k/(k+1)=1/2^{s}\) that holds for \(k=1/(2^{s}-1)>1\). But when \(k>1\) the \(\partial \varOmega _{k}\) is the ellipse.

It is of interest to study the common relations between \(H(s)\) and \(\partial \varOmega _{k}\) for any \(0\le k<\infty \) and \(0< s\le 1\). The detailed investigation of geometric behavior of \(\partial \varOmega _{k}\) and \(H(s)\) provides a following relations:

where

For the case \(0\le k\le 1/2^{s}\) a domain bounded by \(H(s)\) is contained in a domain bounded by \(\partial \varOmega _{k}\), and for \(k\ge 1/(2^{s}-1)\) conversely, the ellipse bounded by \(\partial \varOmega _{k}\) is contained in a domain bounded by a hyperbola \(H(s)\). Let us denote by \(\mathbb{H}(s)\) a domain bounded by hyperbola \(H(s)\). Thus, we have the following inclusion:

In the remaining cases of k the considered domains have nonempty intersection set (see Fig. 1). These observation will have significant consequences for the further relations between corresponding classes of analytic functions, but also means that the domains bounded by hyperbolas \(H(s)\) (\(0< s\le 1\)) are completely different from the similar domains bounded by the hyperbola \(\partial \varOmega _{k}\).

The mutual location of a hyperbola \(H(s)\) (for \(s=1/2\)) and \(\partial \varOmega _{k}\)

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We also note the obvious difference between \(\mathbb{H}(s)\) and other domains contained in a right halfplane: disks, halfplanes and angular domains, that have appeared in the investigations of various subclasses of analytic functions.

From the above reasons we wish to find an analytic function that map the unit disk onto a domain bounded by the hyperbola \(H(s)\), define and investigate a subfamily of \(\mathcal{P}\) related to those domains. As a natural extension of that investigation we will define a subfamily \(\mathcal{ST}_{\mathrm{hpl}}(s)\) of the starlike functions for which \(zf'/f\) is contained in a domain bounded by the hyperbola \(H(s)\) (and related class \(\mathcal{CV}_{\mathrm{hpl}}(s)\) of convex functions). It is also natural to try to give a sufficient condition for the function \(f\in \mathcal{H}\) to be a member of the class \(\mathcal{ST}_{\mathrm{hpl}}(s)\) (or \(\mathcal{CV}_{\mathrm{hpl}}(s)\), resp.) and solve several extremal problems.

In order to achieve our aim we recall some definitions and preliminary results from the geometric theory of analytic functions. By \(\mathcal{B}\) we denote the class of analytic self-mappings of the unit disk that send the origin onto the origin, i.e.

The class \(\mathcal{B}\) is known as the class of Schwarz functions.

We say that f is subordinate to F in \(\mathbb{D}\), written \(f(z)\prec F(z)\) (or \(f\prec F\)), if there exists a function \(\omega \in \mathcal{B}\) such that \(f(z)=F(\omega (z))\) (\(z\in \mathbb{D}\)) see, for example [2, 3]. From that definition it is easy to show that if F is univalent in \(\mathbb{D}\) then \(f\prec F\) holds if and only, if \(f(0)=F(0)\), and \(f(\mathbb{D}) \subset F(\mathbb{D})\).

Let us denote by \(\mathcal{Q}\) the class of functions f that are analytic and injective on \(\overline{\mathbb{D}}\setminus E(f)\), where

and that are such that

Lemma 1.1

([8])

Let \(q\in \mathcal{Q}\) with \(q(0)=1\) and let \(p(z)=1+p_{1}z+\cdots \) be analytic in \(\mathbb{D}\) with \(p(z)\neq 1\). If \(p\nprec q\) in \(\mathbb{D}\) then there exist points \(z_{0}\in \mathbb{D}\) and \(\zeta \in \partial \mathbb{D}\setminus E(q)\) and there exist a real number \(m\geq 1\) for which

This section provides a detailed exposition of an analytic function that maps the unit disk onto a domain bounded by a hyperbola \(H(s)\) and contained in a right halfplane. Let

where the branch of the logarithm is determined by \(\mathfrak {q}_{s}(0)=1\). It is easy to check that

The function \(\mathfrak {q}_{s}\) maps the unit disk onto a domain bounded by a right branch of a hyperbola

Indeed, setting \(z=\mathrm {e}^{\mathrm {i}\theta }\) (\(\theta \in (0,2 \pi ) \)) we obtain

Thus, we have

Setting \(\varphi =(\theta - \pi )s/2\), after simple computation, we obtain

The function

attains its minimal value when \(\theta =\pi \), i.e., we have

Also, we note that \(\Re {\mathfrak {q}_{s}(\mathrm {e}^{\mathrm {i}\theta } )}\to \infty \) for \(\theta \to 0\). From (2.1) we obtain that the boundary \(\mathfrak {q}_{s}(\mathrm {e}^{\mathrm {i}\theta } )\) satisfies the equality \(1/\rho = (2\sin \frac{\theta }{2} )^{s}= (2\cos \frac{ \varphi }{s} )^{s}\). Additionally, we have \(\mathfrak {q}_{s}(0)=1\), therefore a right branch of a hyperbola

is a boundary of a domain \(\mathfrak {q}_{s}(\mathbb{D})\), which we denote by \(\mathbb{H}(s)\). The domain \(\mathbb{H}(s)\) is symmetric about the real axis, starlike with respect to the point \(\mathfrak {q}_{s}(0)=1\), and it satisfies \(\mathfrak {q}_{s}'(0)= s >0\). Also, the hyperbola \({H}(s)=\partial \mathbb{H}(s)\) has a vertex at \((2^{-s}, 0)\) and the slant asymptotes \(v=\pm \tan ( \pi s/2)u\) (see Fig. 2).

The image of \(\mathfrak {q}_{s}(z)=\frac{1}{ (1-z )^{s}}\) (\(s=1/2\))

Full size image

A special and interesting case of \(H(s)\) is that for \(s=1/2\). We then obtain \(\mathfrak{q}_{1/2}(z)=\frac{1}{\sqrt{1-z}}\), and the hyperbola reduces to that given by the relation

Lemma 2.1

The function \(\mathfrak {q}_{s}\) is convex univalent in \(\mathbb{D}\), moreover \(g(z)=(\mathfrak {q}_{s}(z)-1)/s\in \mathcal{CV}((1-s)/2)\). Also, if \(|z|=r<1\), then

Proof

We first observe that \(\mathfrak {q}_{s}\) is convex by a convexity property of a set \(\mathfrak {q}_{s}(\mathbb{D})\). To deduce the univalence, we use the results on univalence of some integrals from [10]. In order to prove (2.2) we consider the function \(Q(\theta )\) for \(\theta \in [0,2\pi )\), where

Since \(Q(\theta )\) attains its minimum at \(\theta =\pi \) and maximum at \(\theta =0\), (2.2) follows.

Let us consider \(g(z)= (\mathfrak {q}_{s}(z)-1)/s \) (\(z\in \mathbb{D} \)). Then we have

so that \(g\in \mathcal{CV}((1-s)/2)\), as asserted. □

Definition 2.2

By \(\mathcal{P}(\mathfrak {q}_{s})\) we denote a class of all analytic functions p such that \(p(0)=1\) and \(p(z)\prec \mathfrak {q}_{s}(z)\), \(z\in \mathbb{D}\), that is,

Theorem 2.3

If \(p\in \mathcal{P}(\mathfrak {q}_{s})\), then

and

or

Conversely, if p is analytic with \(p(0)=1\), \(\vert \operatorname{Arg}\{ p(z) \} \vert <(\pi s)/2\) and p satisfies (2.4) (or one of the conditions of (2.5)), then \(p\prec \mathfrak {q}_{s}\) in \(\mathbb{D}\).

Proof

The subordination \(p\prec \mathfrak {q}_{s}\) with \(p(0)=\mathfrak {q}_{s}(0)\), and the geometric properties of \(\mathfrak {q}_{s}(\mathbb{D})\) from Sect. 1, yield (2.3). We next claim that (2.4) is satisfied. Repeated application of the properties of subordination \(p\prec \mathfrak {q}_{s}\) shows that there exists \(\omega \in \mathcal{B}\), such that

which is equivalent to

and the assertion (2.4) now follows. Squaring (2.4) we may rewrite it as

or

which is nothing but the second relation of (2.5). For the proof of the first relation of (2.5) we rewrite (2.4) as

or equivalently

which immediately becomes

and, from this, the desired conclusion follows.

Conversely, if for p is analytic in \(\mathbb{D}\) with \(p(0)=1\) and the condition (2.4) (or one of the conditions of (2.5)) is satisfied, then we easily show that \(p=\rho \mathrm {e}^{\mathrm {i}\theta }\) lies in a domain \(\mathbb{H}(s)=\mathfrak {q}_{s}(\mathbb{D})\), which completes the proof. □

Theorem 2.4

If \(p\in \mathcal{P}\) satisfies

then

Proof

Suppose, on the contrary, that \(p(z)\nprec \mathfrak {q}_{s}(z)\) on \(\mathbb{D}\). Then, by \(p(0)=\mathfrak {q}_{s}(0)=1\), and by Lemma 1.1, there exist \(z_{0}\in \mathbb{D}\) and \(\zeta _{0} \in \partial \mathbb{D}\) (\(\zeta _{0}\neq 1 \)) such that

Hence

which contradicts the assumption (2.6), and therefore \(p\prec \mathfrak {q}_{s}\) on \(\mathbb{D}\). □

In this section we introduce and examine the notion of starlikeness and convexity related to the domains bounded by the hyperbolas \(H(s)\). We indicate how the properties of the domains \(\mathbb{H}(s)\) and class \(\mathcal{P}(\mathfrak {q}_{s})\) influence on the properties of a newly introduced classes.

Definition 3.1

Let \(\mathcal{ST}_{\mathrm{hpl}}(s)\) denote the subfamily of \(\mathcal{S}\) consisting of the functions f satisfying the condition

and let \(\mathcal{CV}_{\mathrm{hpl}}(s)\) be a class of analytic functions f such that

Geometrically, the condition (3.1) and (3.2) means that the expression \(zf'(z)/f(z)\) (or \(1+zf''(z)/f'(z)\), resp.) lies in a domain \(\mathbb{H}(s)\). Since \(\mathbb{H}(s)=\mathfrak {q}_{s}(\mathbb{D})\) is contained in a right halfplane, we deduce that \(\mathcal{ST}_{\mathrm{hpl}}(s)\) is a proper subset of a class of a starlike functions \(\mathcal{ST}\) (and \(\mathcal{CV}_{\mathrm{hpl}}(s)\subset \mathcal{CV}\), resp.). Furthermore, the properties of \(\mathbb{H}(s)\) considered in Sect. 1, yield

The above inclusions additionally give

Also, geometric properties of \(\varOmega _{k}\) and \(\mathbb{H}(s)\) given by (1.1), (1.2), (1.3) and also (1.4) imply

Moreover

and

Since \(\mathfrak {q}_{s}\) is the extremal function in \(\mathcal{P}(\mathfrak {q}_{s})\), the obvious integral representation of \(\mathcal{ST}_{\mathrm{hpl}}(s)\) and \(\mathcal{CV} _{\mathrm{hpl}}(s)\) immediately follows.

Theorem 3.2

A function f is in the class \(\mathcal{ST}_{\mathrm{hpl}}(s)\) if, and only if there exists p such that \(p\prec \mathfrak {q}_{s}\), and

Theorem 3.3

A function f is in the class \(\mathcal{CV}_{\mathrm{hpl}}(s)\) if, and only if there exists p such that \(p\prec \mathfrak {q}_{s}\), and

Suppose that \(\varPhi _{s,n}\in \mathcal{ST}_{\mathrm{hpl}}(s)\) is such that

Then the functions \(\varPhi _{s,n}(z)\) are of the form

and these are extremal functions for different problems in the class \(\mathcal{ST}_{\mathrm{hpl}}(s)\). For instance

The special case \(s=1/2\) gives

and

Also, suppose that \(K_{s,n}\in \mathcal{CV}_{\mathrm{hpl}}(s)\) is such that

Then the functions \(K_{s,n}(z)\) are of the form

and the \(K_{s,n}\) are extremal functions for various problems in the class \(\mathcal{CV}_{\mathrm{hpl}}(s)\). The most interesting special cases are for \(n=1\)

and for \(s=1/2\), \(n=1\)

From Theorems 3.2 and 3.3 we obtain the following corollary.

Corollary 3.4

If \(f\in \mathcal{CV}(1-s/2)\), then \(f'\prec \mathfrak {q}_{s}\) in \(\mathbb{D}\). Thus f is univalent in \(\mathbb{D}\). The functions H and G defined by

belong to \(\mathcal{ST}_{\mathrm{hpl}}(s)\) and \(\mathcal{CV}_{\mathrm{hpl}}(s)\), respectively.

Examples

Defined classes \(\mathcal{ST}_{\mathrm{hpl}}(s)\) and \(\mathcal{CV}_{\mathrm{hpl}}(s)\) are nonempty. The integral representation given in Theorems 3.2 and 3.3 provides various examples of functions of those classes. For example, if \(q_{1}(z) = 1+B_{1}z\) with \(0< B_{1}<1-2^{-s}\), then \(q_{1}\prec \mathfrak {q}_{s}\). Hence \(f_{1}(z) = z\exp (B_{1}z) \in \mathcal{ST}_{\mathrm{hpl}}(s)\), and \(g_{1}(z) = (e^{B_{1}z}-1)/B_{1} \in \mathcal{CV}_{\mathrm{hpl}}(s)\) (for \(0< B_{1}<1-2^{-s}\)), respectively. Since

for \(0<|B_{2}|\le s/(2-s)\), \(0<|B_{3}|\le s/(4-s)\) are starlike of order \((1-s/2)\), applying Corollary 3.4 we see that the appropriate functions

belong to the class \(\mathcal{ST}_{\mathrm{hpl}}(s)\), and the corresponding functions will be elements of \(\mathcal{CV}_{\mathrm{hpl}}(s)\).

Theorem 3.5

The function \(\varPhi _{s,n}(z)\), defined by (3.4), is normalized, univalent and convex in one direction in \(\mathbb{D}\) for \(n=1,2, \ldots , \lfloor {(2^{1-s}+1)/s\rfloor }\), and convex for \(n=1,2, \ldots , \lfloor {2^{1-s}/s}\rfloor \), where \(\lfloor {\cdot }\rfloor \) is the floor function.

Proof

It is a simple matter to check that \(\varPhi _{s,n}(z)\) is normalized by \(\varPhi _{s,n}(0)=\varPhi '_{s,n}(0)-1=0\). By a definition (3.3) of \(\varPhi _{s,n}(z)\), we have

Hence

and from the above we see that

In order to get univalence, it is convenient to use the result by Umezawa [12], with the requirement \(\Re (1+zh''(z)/h'(z) ) > -1/2\) for univalence of h in \(\mathbb{D}\). That condition holds if

which is satisfied for \(n=1,2, \ldots , \lfloor {(2^{1-s}+1)/s \rfloor }\). We recall that the same condition gives the convexity in one direction. Furthermore, \(\varPhi _{s,n}(z)\) is convex in \(\mathbb{D} \) if \(\Re (1+z\varPhi _{s,n}''(z)/\varPhi _{s,n}'(z) ) >0\) holds for \(n\le 2^{1-s}/s\), i.e. for \(n=1,2, \ldots , \lfloor {2^{1-s}/s} \rfloor \). □

We have the following from the results in [7], and Lemma 2.1.

Corollary 3.6

If \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\) and \(|z|=r<1\), then

Equality holds for some \(z_{0}\neq 0\) if and only if f is a rotation of \(\varPhi _{s}\), where \(\varPhi _{s}\) is given by (3.5).

Also, if \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\), then either f is a rotation of \(\varPhi _{s}\) or

Here \(-\varPhi _{s}(-1)\) is understood to be the limit of \(-\varPhi _{s}(-r)\) as r tends to 1.

Similar results hold for functions of \(\mathcal{CV}_{\mathrm{hpl}}(s)\).

Corollary 3.7

If \(f\in \mathcal{CV}_{\mathrm{hpl}}(s)\) and \(|z|=r<1\), then

Equality holds for some \(z_{0}\neq 0\) if and only if f is a rotation of \(K_{s}\), where \(K_{s}\) is given by (3.7).

Also, if \(f\in \mathcal{CV}_{\mathrm{hpl}}(s)\), then either f is a rotation of \(K_{s}\) or

Here \(-K_{s}(-1)\) is understood to be the limit of \(-K_{s}(-r)\) as r tends to 1.

Theorem 3.8

Let \(r_{0}\) denote the positive root of the equation

If \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\), then f is convex in the disk \(|z|< r_{0}\).

Proof

Let \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\). Then from Definition 3.1 we obtain

where \(w\in \mathcal{B}\). Application of Lemma 2.1 and the well-known inequality for Schwarz functions \(|w'(z)|\le (1-|w(z)|^{2})/(1-|z|^{2})\) gives

The function \(\psi (r)= \frac{1}{ (1+r )^{s}}-\frac{rs}{1-r}\), where \(r=|z| \in [0,1)\) is decreasing in \([0,1)\) with \(\psi (0)=1\) and \(\psi (1^{-})=- \infty \). Thus, there exists a unique \(r_{0}\in (0,1)\) such that \(\psi (r_{0})=0\), and for \(0\le r< r_{0}\) we have \(0< \psi (r)\le 1\). This inequality is equivalent to

which is satisfied for \(|z|< r_{0}\), where \(r_{0}\) is the only real positive root of (3.8). □

Theorem 3.9

If \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\), then there exists \(\alpha >1\) such that

Proof

Let \(f\in \mathcal{ST}_{\mathrm{hpl}}(s)\). Then, from Definition 3.1, we obtain

where \(w\in \mathcal{B}\). By Lemma 2.1 and the inequality for Schwarz functions it follows that

The function \(g(r)= \frac{1}{ (1-r )^{s}}+\frac{rs}{1-r}\), where \(r=|z| \in (0,1)\), is increasing in \((0,1)\) and \(g(0)=1\). Thus for \(z\in \mathbb{D}\) we have \(g(|z|)>1\). □

Acknowledgements

The authors thank the editor and the anonymous referees for constructive and pertinent suggestions.

Availability of data and materials

Not applicable.

Funding

This work was partially supported by the Centre for Innovation and Transfer of Natural Sciences and Engineering Knowledge, Faculty of Mathematics and Natural Sciences, University of Rzeszow.

Affiliations

1. University of Rzeszow, Rzeszów, Poland
   • S. Kanas
2. Department of Mathematics, Payame Noor University (PNU), P.O. Box, 19395-3697, Tehran, Iran
   • V. S. Masih
3. Department of Mathematics, Faculty of Science, Urmia University, Urmia, Iran
   • A. Ebadian

Contributions

Each of the authors contributed to each part of this study equally, all authors read and approved the final manuscript.

Corresponding author

Correspondence to S. Kanas.

Competing interests

The authors declare that they have no competing interests.

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