Starlike and convexity properties of q-Bessel-Struve functions

Karima M. Oraby; Zeinab S. I. Mansour

doi:10.1515/dema-2022-0004

Article Open Access

Starlike and convexity properties of q-Bessel-Struve functions

Karima M. Oraby and Zeinab S. I. Mansour

Published/Copyright: April 19, 2022

Published by

Become an author with De Gruyter Brill

Author Information Explore this Subject

From the journal Demonstratio Mathematica Volume 55 Issue 1

Abstract

This paper introduces three different normalization associated with the second and third q-Bessel-Struve functions. We use Hadamard factorizations to determine the radii of starlike and convexity of these functions.

Keywords: starlike; convexity; q-Bessel-Struve functions

MSC 2010: 30C45; 30C15; 33D15

1 Introduction

The Struve functions play a major role in several applications of a wide variety of physical and mathematical problems, for example, in water-wave and surface-wave problems [1], as well as in problems on unsteady aerodynamics [2]. Oraby and Mansour [3] introduced the q-Struve-Bessel functions H ν ( k ) ( z ; q 2 ) , k = 1 , 2 , 3 , defined by

(1.1) H ν ( 1 ) ( z ∣ q 2 ) = ∑ k = 0 ∞ ( − 1 ) k Γ q 2 k + 3 2 Γ q 2 ( k + ν + 3 / 2 ) z 1 + q 2 k + ν + 1 , ∣ z ∣ < 1 1 − q , H ν ( 2 ) ( z ∣ q 2 ) = ∑ k = 0 ∞ ( − 1 ) k q 2 k 2 + 2 k ν + 2 k Γ q 2 k + 3 2 Γ q 2 ( k + ν + 3 / 2 ) z 1 + q 2 k + ν + 1 , z ∈ C ,

(1.2) H ν ( 3 ) ( z ; q 2 ) = ∑ k = 0 ∞ ( − 1 ) k q k 2 + k Γ q 2 k + 3 2 Γ q 2 ( k + ν + 3 / 2 ) z 1 + q 2 k + ν + 1 , z ∈ C ,

we follow [4] for the definitions of q-shifted factorial, q-gamma function, the q-binomial coefficients, Jackson q-difference and q-integral operators and q-numbers.

The functions H ν ( k ) ( z ; q 2 ) , ( k = 1 , 2 , 3 ) , are q-analogues of the classical Struve function [5, p. 328]

H ν ( z ) = ∑ n = 0 ∞ ( − 1 ) n ( x / 2 ) 2 n + ν + 1 Γ n + 3 2 Γ n + ν + 3 2 .

Since

lim q → 1 − H ν ( k ) ( z ∣ q 2 ) = lim q → 1 − H ν ( 3 ) ( z ; q 2 ) = H ν ( z ) , k = 1 , 2 .

One can see that the functions z − ν − 1 H ν ( k ) ( z ; q 2 ) , ( k = 2 , 3 ) are entire functions of order zero, then they have infinitely many zeros. According to Hadamard factorization theorem [6]

(1.3) z − ν − 1 H ν ( 3 ) ( z ; q 2 ) = ( 1 + q ) − ν − 1 Γ q 2 ( 3 / 2 ) Γ q 2 ν + 3 2 ∏ n = 1 ∞ 1 − z 2 z n , ν 2

and

(1.4) z − ν − 1 H ν ( 2 ) ( z ∣ q 2 ) = ( 1 + q ) − ν − 1 Γ q 2 ( 3 / 2 ) Γ q 2 ν + 3 2 ∏ n = 1 ∞ 1 − z 2 z ˜ n , ν 2 ,

where z n , ν and z ˜ n , ν denote the nth positive zero of z − ν − 1 H ν ( 3 ) ( z ; q 2 ) and z − ν − 1 H ν ( 2 ) ( z ∣ q 2 ) , respectively. Oraby and Mansour [3] proved that for ν > − 1 2 , the functions z − ν − 1 H ν ( k ) ( z ; q 2 ) , ( k = 2 , 3 ) have only real simple zeros.

The geometric properties of the classical Struve function H ν are in [7,8,9, 10]. On the other hand, the interested reader may refer to [8,11,12, 13,14,15, 16,17,18, 19,20] to study the radii of univalence, starlike, and convexity for hypergeometric functions, Bessel, Struve, and Lommel functions of the first kind. Moreover, Ismail et al. [21] introduced and investigated the generalized class of starlike functions by using the q-difference operator. Many authors have investigated the geometric properties of some q-special functions (like second and third Jackson q-Bessel functions and q-Mittag-Leffler functions), see [22,23, 24,25]. Motivated by the mentioned work, in this paper, Hadamard factorization of q-Struve-Bessel functions z − ν − 1 H ν ( 2 ) ( z ∣ q 2 ) and z − ν − 1 H ν ( 3 ) ( z ; q 2 ) plays an important role in deriving the main results, as Hadamard factorization of Struve function was obtained in [26]. We consider the approach of Baricz et al. to investigate the radii of starlikeness and convexity of the q-Bessel-Struve functions.

2 Preliminaries

Let D r be the open disk { z ∈ C : ∣ z ∣ < r } with radius r > 0 and let D 1 = D . By A , we mean the class of analytic functions f : D r → C ,

(2.1) f ( z ) = z + ∑ n ≥ 2 a n z n ,

which satisfy the normalization conditions f ( 0 ) = f ′ ( 0 ) − 1 = 0 . By S , we mean the class of functions belonging to A which are univalent in D r . Let S ∗ ( α ) be the subclass of S consisting of functions which are starlike of order α in D r , where α ∈ [ 0 , 1 ) , see [17]. That is,

S ∗ ( α ) = f ∈ S : ℜ 1 + z f ′ ( z ) f ( z ) > α for all z ∈ D r ,

and we adopt the convention S ∗ = S ∗ ( 0 ) . The real number

r α ∗ ( f ) = sup r > 0 : ℜ 1 + z f ′ ( z ) f ( z ) > α for all z ∈ D r

is called the radius of starlikeness of the function f . Note that r ∗ ( f ) = r 0 ∗ ( f ) is the largest radius such that the image region f ( D r ∗ ( f ) ) is a starlike domain with respect to the origin. For 0 ≤ α < 1 , the class of convex functions is defined by

K = f ∈ S : ℜ 1 + z f ″ ( z ) f ′ ( z ) > α for all z ∈ D r .

We note that the convex functions do not need to be normalized, that is, the definition of K is also valid for a non-normalized analytic function f : D r → C with the property f ′ ( 0 ) ≠ 0 . The radius of convexity of an analytic locally univalent function f : C → C is defined by

r α c ( f ) ≔ sup r > 0 : ℜ 1 + z f ″ ( z ) f ′ ( z ) > α for all z ∈ D r .

We note that r c ( f ) = r 0 c ( f ) is the largest radius such that the image region f ( D r c ( f ) ) is a convex domain. We refer to [27] for more details about starlike and convex functions.

For every q , 0 < q < 1 , Ismail et al. [21] introduced the class S q ∗ as a generalization of starlike functions S ∗ by replacing the usual derivative by the q-difference derivative D q and the right-half plane by a suitable domain. They defined the class S q ∗ as follows:

Definition 2.1

A function f ∈ A is in the class S q ∗ if

f ( 0 ) = 0 , f ′ ( 0 ) = 1 , and z f ( z ) ( D q f ) ( z ) − 1 1 − q ≤ 1 1 − q .

In [21], it is proved that

(2.2) S ∗ = ⋂ 0 < q < 1 S q ∗ .

In [28], the authors gave a generalization of starlike functions of order α , α ∈ [ 0 , 1 ) . They defined the class S q ∗ ( α ) by

S q ∗ ( α ) = f ∈ S : z f ( z ) ( D q f ) ( z ) − α 1 − α − 1 1 − q ≤ 1 1 − q , for all z ∈ D .

In particular, when α = 0 , the class S q ∗ ( α ) coincides with the class S q ∗ = S q ∗ ( 0 ) . In [28], it is proved that

(2.3) S ∗ ( α ) = ⋂ 0 < q < 1 S q ∗ ( α ) .

Also, we recall that a real entire function ψ belongs to the Laguerre-Pólya class ℒP if it can be represented in the form

ψ ( x ) = c x m e − a x 2 + b x ∏ n ≥ 1 1 + x x n e − x x n ,

where c , b , x n ∈ R , a ≥ 0 , m ∈ N 0 , and ∑ n ≥ 1 x n − 2 < ∞ . We note that ℒP is the complement of the space of polynomials with only real zeros in the topology induced by the uniform convergence on the compact sets of the complex plane of polynomials with only real zeros. For more details on the class ℒP , we refer to [29].

Since H ν ( 2 ) ( z ∣ q 2 ) and H ν ( 3 ) ( z ; q 2 ) do not belong to A , we perform some natural normalizations as in [9,23]. For ∣ ν ∣ < 1 2 , we associate with H ν ( 3 ) ( z ; q 2 ) the functions:

f ν ( 3 ) ( z ; q 2 ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 H ν ( z ; q 2 ) 1 ν + 1 , ν ≠ − 1 , g ν ( 3 ) ( z ; q 2 ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − ν H ν ( z ; q 2 ) , ζ ν ( 3 ) ( z ; q 2 ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z 1 − ν 2 H ν ( z ; q 2 ) .

Similarly, we associate with H ν ( 2 ) ( z ∣ q 2 ) the functions:

f ν ( 2 ) ( z ; q 2 ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 H ν ( 2 ) ( z ∣ q 2 ) 1 ν + 1 , ν ≠ − 1 , g ν ( 2 ) ( z ; q 2 ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − ν H ν ( 2 ) ( z ∣ q 2 ) , ζ ν ( 2 ) ( z ; q 2 ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z 1 − ν 2 H ν ( 2 ) ( z ∣ q 2 ) .

Clearly, the functions f ν ( k ) ( z ; q 2 ) , g ν ( k ) ( z ; q 2 ) , and ζ ν ( k ) ( z ; q 2 ) , k = 2 , 3 , belong to the class A . We note here that

f ν ( 2 ) ( z ; q 2 ) = exp 1 ν + 1 Log ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 H ν ( 2 ) ( z ∣ q 2 ) , f ν ( 3 ) ( z ; q 2 ) = exp 1 ν + 1 Log ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 H ν ( 3 ) ( z ; q 2 ) ,

where Log represents the principal branch of the logarithm function and every many-valued function considered in this paper is taken with the principle branch.

The following lemma plays an essential role in proving our main results. We will use H ν ( z ; q 2 ) , H ν ′ ( z ; q 2 ) , and H ν ″ ( z ; q 2 ) instead of H ν ( 3 ) ( z ; q 2 ) , d H ν ( 3 ) ( z ; q 2 ) d z , and d 2 H ν ( 3 ) ( z ; q 2 ) d z 2 , respectively.

Lemma 2.2

Between any two consecutive roots of the function H ν ( z ; q 2 ) , the function H ν ′ ( z ; q 2 ) has precisely one zero when ∣ ν ∣ < 1 2 .

Proof

Since the zeros of H ν ( z ; q 2 ) are real, it follows that

ℋ ν ( z ; q 2 ) = ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − ν − 1 H ν ( z ; q 2 )

is in the Laguerre-Pólya class of real entire functions. From (1.3), we have

ℋ ν ( z ; q 2 ) = ∏ n = 1 ∞ 1 − z 2 h ν , n 2 ,

where h n , ν denote the nth positive zero of z − ν − 1 H ν ( 3 ) ( z ; q 2 ) . It follows that it satisfies the Laguerre inequality, see [30]

(2.4) ( ℋ ν ( n ) ( z ; q 2 ) ) 2 − ( ℋ ν ( z ; q 2 ) ) ( n − 1 ) ( ℋ ν ( z ; q 2 ) ) ( n + 1 ) > 0 ,

where n ∈ N , ∣ ν ∣ < 1 2 , and z ∈ R . On the other hand,

ℋ ν ′ ( z ; q 2 ) = C ν ( q ) z − ν − 2 [ z H ν ′ ( z ; q 2 ) − ( ν + 1 ) H ν ( z ; q 2 ) ] , ℋ ν ″ ( z ; q 2 ) = C ν ( q ) z − ν − 3 [ z 2 H ν ″ ( z ; q 2 ) − 2 ( ν + 1 ) z H ν ′ ( z ; q 2 ) + ( ν + 1 ) ( ν + 2 ) H ν ( z ; q 2 ) ] ,

where C ν ( q ) ≔ ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 . Thus, the Laguerre inequality (2.4) for n = 1 is equivalent to

( C ν ( q ) ) 2 z − 2 ν − 4 [ ( z H ν ′ ( z ; q 2 ) ) 2 − z 2 H ν ( z ; q 2 ) H ν ″ ( z ; q 2 ) − ( ν + 1 ) H ν 2 ( z ; q 2 ) ] > 0 ,

which implies that

( H ν ′ ( z ; q 2 ) ) 2 − H ν ( z ; q 2 ) H ν ″ ( z ; q 2 ) > ( ν + 1 ) H ν 2 ( z ; q 2 ) / z 2 > 0 ,

for ∣ ν ∣ < 1 2 and z ≠ 0 . Consequently, the function H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) is strictly decreasing on each interval ( h ν , n − 1 , h ν , n ) , n ∈ N (note that h ν , 0 = 0 ). Since the zeros h ν , n of the function H ν ( z ; q 2 ) are simple, the function H ν ′ ( z ; q 2 ) does not vanish in h ν , n , n ∈ N . On the other hand, for fixed n ∈ N ,

lim x → h ν , n − H ν ′ ( x ; q 2 ) H ν ( x ; q 2 ) = − ∞ , lim x → h ν , n + H ν ′ ( x ; q 2 ) H ν ( x ; q 2 ) = ∞ .

Summarizing for arbitrary n ∈ N the graph of the restriction of the function H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) to each interval ( h ν , n − 1 , h ν , n ) intersects the horizontal axis only once, and the abscissa of this intersection point is exactly h ν , n ′ . Thus, we proved that when ∣ ν ∣ < 1 2 the positive zeros of H ν ′ ( z ; q 2 ) and H ν ( z ; q 2 ) are interlacing.□

Lemma 2.3

Between any two consecutive roots, the function H ν ( 2 ) ′ ( z ∣ q 2 ) has precisely one zero when ∣ ν ∣ < 1 2 .

In the following sections, we aim to determine the radii of starlikeness and convexity of the above mentioned normalized functions precisely by using their Hadamard factorization. This generalizes some known results for Struve functions [9].

3 The radii of starlike of q-Bessel-Struve functions

The following result will be used in the proof of Theorem 1.

Lemma 3.1

[17] Let f be as in (2.1). If

∑ n = 2 ∞ ( n − α ) ∣ a n ∣ ≤ 1 − α ,

then the function f is in the class S ∗ ( α ) .

Theorem 1

Let q ∈ ( 0 , 1 ) and α ∈ [ 0 , 1 ) . Assume that

(3.1) ln 1 q + q − 1 2 ln q < ν < ln 1 q + 2 q − 2 2 ln q .

If the inequality

(3.2) α ≥ 2 q 2 ( 1 − q ) 2 + ( 1 − q 2 ν + 1 ) 2 − 4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 q 2 ( 1 − q ) 2 + ( 1 − q 2 ν + 1 ) 2 − 3 q ( 1 − q ) ( 1 − q 2 ν + 1 )

holds, then the normalized q-Struve-Bessel function ζ ν ( 3 ) ( z ; q 2 ) is a starlike of order α .

Proof

By Theorem 3.1, it is enough to show that the following inequality

(3.3) k 1 = ∑ n ≥ 2 ( n − α ) ( − 1 ) n − 1 q n ( n − 1 ) ( 1 − q ) 2 n − 1 ( 1 − q 2 ν + 1 ) ( q ; q 2 ) n ( q 2 ν + 1 ; q 2 ) n ≤ 1 − α

holds under the hypothesis. It is clear that ( q ; q 2 ) n ≥ ( 1 − q ) n and ( q 2 ν + 1 ; q 2 ) n ≥ ( 1 − q 2 ν + 1 ) n , for ν > − 1 2 and (3.1) implies that

1 < 1 − q 2 ν + 1 q ( 1 − q ) < 2 .

Then

k 1 ≤ ∑ n ≥ 2 ( n − α ) q n − 1 ( 1 − q ) n − 1 ( 1 − q 2 ν + 1 ) n − 1 = ∑ n ≥ 2 n q ( 1 − q ) ( 1 − q 2 ν + 1 ) n − 1 − α ∑ n ≥ 2 q ( 1 − q ) ( 1 − q 2 ν + 1 ) n − 1 = 2 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − q 2 ( 1 − q ) 2 [ 1 − q 2 ν + 1 − q ( 1 − q ) ] 2 − α q ( 1 − q ) 1 − q 2 ν + 1 − q ( 1 − q ) .

Here, we used the following series sums

(3.4) ∑ n ≥ 2 r n − 1 = r 1 − r

and

(3.5) ∑ n ≥ 2 n r n − 1 = r ( 2 − r ) ( 1 − r ) 2 .

The inequality (3.2) implies that k 1 ≤ 1 − α and so the function ζ ν ( 3 ) ( z ; q 2 ) is starlike of order α .□

Remark 3.2

It is worth noting that (3.1) implies that ∣ ν ∣ < 1 2 .

Theorem 2

Let q ∈ ( 0 , 1 ) and α ∈ [ 0 , 1 ) . Assume that

(3.6) ln ( 2 q − q 2 ) − 2 ln q < ν < ln ( 3 q − 2 q 2 ) − 2 ln q .

If the inequality

(3.7) α ≥ 2 q 4 ν + 2 ( 1 − q ) 2 + ( 1 − q 2 ν + 1 ) 2 − 4 q 2 ν + 1 ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 q 4 ν + 2 ( 1 − q ) 2 + ( 1 − q 2 ν + 1 ) 2 − 3 q 2 ν + 1 ( 1 − q ) ( 1 − q 2 ν + 1 )

holds, then the normalized q-Struve-Bessel function ζ ν ( 2 ) ( z ; q 2 ) is a starlike of order α .

Remark 3.3

It is worth noting that (3.6) implies that

q 2 ν < 1 − q 2 ν + 1 q ( 1 − q ) < 2 q 2 ν

and ∣ ν ∣ < 1 2 .

Theorem 3

Let ∣ ν ∣ < 1 2 and α ∈ [ 0 , 1 ) . Then the following assertions hold

r α ∗ ( f ν ( 3 ) ( z ; q 2 ) ) = x ν , α , 1 , where x ν , α , 1 is the smallest positive root of the equation
r H ν ′ ( r ; q 2 ) − α ( ν + 1 ) H ν ( r ; q 2 ) = 0 .
r α ∗ ( g ν ( 3 ) ( z ; q 2 ) ) = y ν , α , 1 , where y ν , α , 1 is the smallest positive root of the equation
r H ν ′ ( r ; q 2 ) − ( α + ν ) H ν ( r ; q 2 ) = 0 .
r α ∗ ( ζ ν ( z ; q 2 ) ) = z ν , α , 1 , where z ν , α , 1 is the smallest positive root of the equation
r H ν ′ ( r ; q 2 ) − ( ν + 2 α − 1 ) H ν ( r ; q 2 ) = 0 .

Proof

From (1.3), we have

(3.8) H ν ( z ; q 2 ) = z ν + 1 ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 ∏ n = 1 ∞ 1 − z 2 h n , ν 2 ( q 2 ) ,

where h ν , n ( q 2 ) stands to the nth positive zero of H ν ( z ; q 2 ) . From (3.8), we have

(3.9) z H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) = ν + 1 − ∑ n ≥ 1 2 z 2 h ν , n 2 ( q 2 ) − z 2 .

It follows that

z f ν ′ ( z ; q 2 ) f ν ( z ; q 2 ) = 1 − 1 ν + 1 ∑ n ≥ 1 2 z 2 h ν , n 2 ( q 2 ) − z 2 .

Similarly, we have

z g ν ′ ( z ; q 2 ) g ν ( z ; q 2 ) = − ν + z H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) = 1 − ∑ n ≥ 1 2 z 2 h ν , n 2 ( q 2 ) − z 2

and

z ζ ν ′ ( z ; q 2 ) ζ ν ( z ; q 2 ) = 1 − ν 2 + z H ν ′ ( z ; q 2 ) 2 H ν ( z ; q 2 ) = 1 − ∑ n ≥ 1 2 z 2 h ν , n 2 ( q 2 ) − z 2 .

On the other hand, it is known that (see [14]) if z ∈ C and β ∈ R are such that β > ∣ z ∣ , then

(3.10) ∣ z ∣ β − ∣ z ∣ ≥ ℜ z β − z .

Consequently, the inequality

∣ z ∣ 2 h ν , n 2 ( q 2 ) − ∣ z ∣ 2 ≥ ℜ z 2 h ν , n 2 ( q 2 ) − z 2

holds for every ∣ ν ∣ < 1 2 , n ∈ N , and ∣ z ∣ < h ν , 1 ( q 2 ) , which in turn implies that

ℜ z f ν ′ ( z ; q 2 ) f ν ( z ; q 2 ) ≥ 1 − 1 ν + 1 ∑ n ≥ 1 2 ∣ z ∣ 2 h ν , n 2 ( q 2 ) − ∣ z ∣ 2 = ∣ z ∣ f ν ′ ( ∣ z ∣ ; q 2 ) f ν ( ∣ z ∣ ; q 2 ) , ℜ z g ν ′ ( z ; q 2 ) g ν ( z ; q 2 ) ≥ 1 − ∑ n ≥ 1 2 ∣ z ∣ 2 h ν , n 2 ( q 2 ) − ∣ z ∣ 2 = ∣ z ∣ g ν ′ ( ∣ z ∣ ; q 2 ) g ν ( ∣ z ∣ ; q 2 ) ,

and

ℜ z ζ ν ′ ( z ; q 2 ) ζ ν ( z ; q 2 ) ≥ 1 − ∑ n ≥ 1 ∣ z ∣ h ν , n 2 ( q 2 ) − ∣ z ∣ 2 = ∣ z ∣ ζ ν ′ ( ∣ z ∣ ; q 2 ) ζ ν ( ∣ z ∣ ; q 2 ) .

Then the minimum principle for harmonic functions [31] implies that

ℜ z f ν ′ ( z ; q 2 ) f ν ( z ; q 2 ) > α if and only if ∣ z ∣ < x ν , α , 1 ,

where x ν , α , 1 is the smallest positive root of

r H ν ′ ( r ; q 2 ) − α ( ν + 1 ) H ν ( r ; q 2 ) = 0 .

Similarly,

ℜ z g ν ′ ( z ; q 2 ) g ν ( z ; q 2 ) > α if and only if ∣ z ∣ < y ν , α , 1 ,

where y ν , α , 1 is the smallest positive root of

r H ν ′ ( r ; q 2 ) − ( α + ν ) H ν ( r ; q 2 ) = 0

and

ℜ z ζ ν ′ ( z ; q 2 ) ζ ν ( z ; q 2 ) > α if and only if ∣ z ∣ < z ν , α , 1 ,

where z ν , α , 1 is the smallest positive root of

r H ν ′ ( r ; q 2 ) − ( ν + 2 α − 1 ) H ν ( r ; q 2 ) = 0 . □

Theorem 4

Let ∣ ν ∣ < 1 2 and α ∈ [ 0 , 1 ) . Then the following assertions hold.

r α ∗ ( f ν ( 2 ) ( z ; q 2 ) ) = x ˜ ν , α , 1 , where x ˜ ν , α , 1 is the smallest positive root of the equation
r H ν ( 2 ) ′ ( r ∣ q 2 ) − α ( ν + 1 ) H ν ( 2 ) ( r ∣ q 2 ) = 0 .
r α ∗ ( g ν ( 2 ) ( z ; q 2 ) ) = y ˜ ν , α , 1 , where y ˜ ν , α , 1 is the smallest positive root of the equation
r H ν ( 2 ) ′ ( r ∣ q 2 ) − ( α + ν ) H ν ( 2 ) ( r ∣ q 2 ) = 0 .
r α ∗ ( ζ ν ( 2 ) ( z ; q 2 ) ) = z ˜ ν , α , 1 , where z ˜ ν , α , 1 is the smallest positive root of the equation
r H ν ( 2 ) ′ ( r ∣ q 2 ) − ( ν + 2 α − 1 ) H ν ( 2 ) ( r ∣ q 2 ) = 0 .

The following theorem gives the upper and lower bounds for the radii of starlikeness of functions seen in the above theorem.

Theorem 5

Let ∣ ν ∣ < 1 2 . Then the following assertions hold

The radius of starlikeness r ∗ ( f ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
(3.11) q − 2 [ 3 ] q [ 2 ν + 3 ] q ν + 3 < ( r ∗ ( f ν ( 3 ) ( z ; q 2 ) ) ) 2 < q − 2 ( ν + 1 ) ( ν + 3 ) [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q ( ν + 3 ) 2 [ 5 ] [ 2 ν + 5 ] q − 2 q 2 ( ν + 5 ) [ 3 ] q [ 2 ν + 3 ] q .
The radius of starlikeness r ∗ ( g ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
(3.12) 1 6 q 2 [ 3 ] q [ 2 ν + 3 ] q < ( r ∗ ( g ν ( 3 ) ( z ; q 2 ) ) ) 2 < 3 [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q q 2 ( 9 [ 5 ] [ 2 ν + 5 ] q − 10 q 2 [ 3 ] q [ 2 ν + 3 ] q ) .
The radius of starlikeness r ∗ ( ζ ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
(3.13) 1 2 q 2 [ 3 ] q [ 2 ν + 3 ] q < r ∗ ( ζ ν ( 3 ) ( z ; q 2 ) ) < [ 5 ] q [ 3 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q 2 q 2 [ 5 ] q [ 2 ν + 5 ] q − 3 q 4 [ 3 ] q [ 2 ν + 3 ] q .

Proof

The radius of starlikness of the function f ν ( 3 ) ( z ; q 2 ) coincides with the radius of the function H ν ( 3 ) ( z ; q 2 ) . From the infinite series representation (1.2), we have
(3.14) H ν ′ ( z ; q 2 ) = 1 1 + q ∑ n = 0 ∞ ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n + ν
and
(3.15) H ν ″ ( z ; q 2 ) = 1 ( 1 + q ) 2 ∑ n = 0 ∞ ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) ( 2 n + ν ) Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n + ν − 1 .
By (1.3), the function H ν ( z ; q 2 ) is in the class ℒP , which is closed under differentiation. Therefore, the function H ν ′ ( z ; q 2 ) is in the class ℒP . On the other hand, see [32], if f is an entire function with non-integer infinite growth, then f has infinitely many zeros or f is a polynomial. Therefore, since the growth order of the real entire function z − ν H ν ′ ( z ; q 2 ) is 1 2 , then H ν ′ ( z ; q 2 ) has infinitely many zeros when ∣ ν ∣ < 1 2 . Now, by applying Hadamard’s theorem [6], we can write the infinite product representation of H ν ′ ( z ; q 2 ) in the form
(3.16) H ν ′ ( z ; q 2 ) = ( ν + 1 ) z ν ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 ∏ n ≥ 1 1 − z 2 h ν , n ′ 2 ,
where h ν , n ′ denotes the nth positive zero of H ν ′ ( z ; q 2 ) . Take the logarithmic derivative of both sides of (3.16) for ∣ z ∣ < h 1 , ν ′ , we obtain
(3.17) z H ν ″ ( z ; q 2 ) H ν ′ ( z ; q 2 ) − ν = − ∑ n ≥ 1 2 z 2 h ν , n ′ 2 − z 2 = − 2 ∑ n ≥ 1 ∑ k ≥ 0 z 2 k + 2 h ν , n ′ 2 k + 2 = − 2 ∑ k ≥ 0 ∑ n ≥ 1 z 2 k + 2 h ν , n ′ 2 k + 2 = − 2 ∑ k ≥ 0 a k + 1 z 2 k + 2 ,
where a k = ∑ n ≥ 1 1 h ν , n ′ 2 k . On the other hand, by using (3.14) and (3.15), we obtain
(3.18) z H ν ″ ( z ; q 2 ) H ν ′ ( z ; q 2 ) = ∑ n ≥ 0 b n z 2 n ∑ n ≥ 0 c n z 2 n ,
where
b n = ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) ( 2 n + ν ) ( 1 + q ) 2 n Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2
and
c n = ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) ( 1 + q ) 2 n Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 .
By comparing the coefficient of (3.17) and (3.18), we obtain
b 0 = ν c 0 , b 1 = ν c 1 − 2 c 0 a 1 ,
and
b n = ν c n − 2 ∑ k = 0 n − 1 n − 1 k a k + 1 c n − k − 1 , n ≥ 2 ,
which gives
a 1 = q 2 ( ν + 3 ) [ 3 ] q [ 2 ν + 3 ] q
and
a 2 = q 4 ( ( ν + 3 ) 2 [ 5 ] q [ 2 ν + 5 ] q − 2 q 2 ( ν + 5 ) [ 3 ] q [ 2 ν + 3 ] q ) ( ν + 1 ) [ 3 ] q 2 [ 5 ] q [ 2 ν + 3 ] q 2 [ 2 ν + 5 ] q .
By using the Euler-Rayleigh inequality [33]
a k − 1 k < h ν , 1 ′ 2 ( q 2 ) < a k a k + 1 ,
for k = 1 , we obtain (3.11).
If we take α = 0 in the part (b) of Theorem 3, then the radius of starlike of order zero of the function g ν ( 3 ) ( z ; q 2 ) is the smallest positive root of the equation
( g ν ( 3 ) ( z ; q 2 ) ) ′ = 0 .
We can see that
(3.19) g ν ′ ( z ; q 2 ) = ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − ν − 1 [ z H ν ′ ( z ; q 2 ) − ν H ν ( z ; q 2 ) ] = ∑ n = 0 ∞ ( − 1 ) n q n 2 + n ( 2 n + 1 ) Γ q 2 3 2 Γ q 2 ν + 3 2 Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n ,
and it is an entire function of order ρ = 1 2 . So, by applying Hadamard’s theorem, we can write the infinite product representation of g ν ′ ( z ; q 2 ) as follows:
(3.20) g ν ′ ( z ; q 2 ) = ∏ n ≥ 1 1 − z 2 β ν , n 2 ( q 2 ) ,
where β ν , n ( q 2 ) denotes the nth positive zero of g ν ′ ( z ; q 2 ) . From [16], for ∣ ν ∣ < 1 2 , the function g ν ′ ( z ; q 2 ) belongs to the Laguerre-Pólya class of entire functions, and the smallest positive zero of g ν ′ ( z ; q 2 ) does not exceed the first positive zero of H ν ( z ; q 2 ) . Logarithmic differentiation of both sides of (3.20) for ∣ z ∣ < β ν , 1 ( q 2 ) gives
(3.21) g ν ( 3 ) ″ ( z ; q 2 ) g ν ( 3 ) ′ ( z ; q 2 ) = ∑ n ≥ 1 − 2 z β ν , n ( q 2 ) − z 2 = − 2 ∑ k ≥ 0 λ k + 1 z 2 k + 1 ,
where λ k = ∑ n ≥ 1 β ν , n − 2 k ( q 2 ) . Also, we have
(3.22) g ν ( 3 ) ″ ( z ; q 2 ) g ν ( 3 ) ′ ( z ; q 2 ) = − 2 ∑ n ≥ 0 c n z 2 n + 1 ∑ n ≥ 0 d n z 2 n ,
where
c n = ( − 1 ) n q n 2 + 3 n + 2 ( 2 n + 3 ) ( 2 n + 2 ) Γ q 2 n + 5 2 Γ q 2 n + ν + 5 2 ( 1 + q ) 2 n + 2
and
d n = ( − 1 ) n q n 2 + n ( 2 n + 1 ) Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 ( 1 + q ) 2 n .
By comparing the coefficient in (3.21) and (3.22), we obtain
d 0 λ 1 = c 0 , and d 0 λ 2 + d 1 λ 1 = c 1 ,
which gives
λ 1 = 6 q 2 [ 3 ] q [ 2 ν + 3 ] q
and
λ 2 = − 20 q 6 [ 3 ] q [ 2 ν + 3 ] q + 18 q 4 [ 5 ] q [ 2 ν + 5 ] q [ 3 ] q 2 [ 2 ν + 3 ] q 2 [ 5 ] q [ 2 ν + 5 ] q .
By using the Euler-Rayleigh inequality
λ k − 1 k < β ν , 1 2 ( q 2 ) < λ k λ k + 1 ,
for k = 1 , we obtain (3.12).
If we take α = 0 in the part (c) of Theorem 3, then the radius of starlike of order zero of the function ζ ν ( 3 ) ( z ; q 2 ) is the smallest positive root of the equation
( ζ ν ( 3 ) ( z ; q 2 ) ) ′ = 0 .
Observe that
(3.23) ζ ν ( 3 ) ′ ( z ; q 2 ) = 1 2 ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − 1 − ν 2 [ z H ν ′ ( z ; q 2 ) + ( 1 − ν ) H ν ( z ; q 2 ) ] = ∑ n = 0 ∞ ( − 1 ) n q n 2 + n ( n + 1 ) Γ q 2 3 2 Γ q 2 ν + 3 2 Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n
is an entire function of order ρ = 1 2 . So, by applying Hadamard’s theorem,
(3.24) ζ ν ( 3 ) ′ ( z ; q 2 ) = ∏ n ≥ 1 1 − z δ ν , n ( q 2 ) ,
where δ ν , n ( q 2 ) denotes the nth positive zero of ζ ν ( 3 ) ′ ( z ; q 2 ) . From [16], for ∣ ν ∣ < 1 2 , the function ζ ν ( 3 ) ′ ( z ; q 2 ) is in the Laguerre-Pólya class of entire functions, and the smallest positive zero of ζ ν ( 3 ) ′ ( z ; q 2 ) does not exceed the first positive zero of H ν ( z ; q 2 ) . By (3.24), we obtain
(3.25) ζ ν ( 3 ) ″ ( z ; q 2 ) ζ ν ( 3 ) ′ ( z ; q 2 ) = − ∑ n ≥ 1 1 δ ν , n − z = − ∑ k ≥ 0 μ k + 1 z k ,
where μ k = ∑ n ≥ 1 δ ν , n − k ( q 2 ) . On the other hand, we have
(3.26) ζ ν ( 3 ) ″ ( z ; q 2 ) ζ ν ( 3 ) ′ ( z ; q 2 ) = − ∑ n ≥ 0 u n z n ∑ n ≥ 0 v n z n ,
where
u n = ( − 1 ) n q n 2 + 3 n + 2 ( n + 1 ) ( n + 2 ) Γ q 2 n + 5 2 Γ q 2 n + ν + 5 2 ( 1 + q ) 2 n + 2
and
v n = ( − 1 ) n q n 2 + n ( n + 1 ) Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 ( 1 + q ) 2 n .
By comparing the coefficient in (3.25) and (3.26), we obtain
v 0 μ 1 = u 0 and v 0 μ 2 + v 1 μ 1 = u 1 ,
it gives
μ 1 = 2 q 2 [ 3 ] q [ 2 ν + 3 ] q
and
μ 2 = − 6 q 6 [ 3 ] q [ 2 ν + 3 ] q + 4 q 4 [ 5 ] q [ 2 ν + 5 ] q [ 3 ] q 2 [ 2 ν + 3 ] q 2 [ 5 ] q [ 2 ν + 5 ] q .
By using the Euler-Rayleigh inequality
μ k − 1 k < δ ν , 1 ( q 2 ) < μ k μ k + 1 ,
for k = 1 , we obtain (3.13).□

Corollary 3.4

Let ∣ ν ∣ < 1 2 . Then the following assertions hold

If 0 < q < 6 11 , then
r ∗ ( f ν ( 3 ) ( z ; q 2 ) ) ∈ 1 q 2 7 [ 3 ] q [ 2 ν + 3 ] q , 1 q 6 [ 5 ] q [ 2 ν + 5 ] q .
If 0 < q < 9 10 , then
r ∗ ( g ν ( 3 ) ( z ; q 2 ) ) ∈ 1 q 1 6 [ 3 ] q [ 2 ν + 3 ] q , 1 q 3 [ 5 ] q [ 2 ν + 5 ] q 9 − 10 q 2 .
If 0 < q < 2 3 , then
r ∗ ( ζ ν ( 3 ) ( z ; q 2 ) ) ∈ 1 2 q 2 [ 3 ] q [ 2 ν + 3 ] q , [ 5 ] q [ 2 ν + 5 ] q q 2 ( 2 − 3 q 2 ) .

Theorem 6

Let ∣ ν ∣ < 1 2 . Then the following assertions hold

The radius of starlikeness r ∗ ( f ν ( 2 ) ( z ; q 2 ) ) satisfies the inequality
[ 3 ] q [ 2 ν + 3 ] q q 2 ν + 4 ( ν + 3 ) < ( r ∗ ( f ν ( 2 ) ( z ; q 2 ) ) ) 2 < ( ν + 1 ) ( ν + 3 ) [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q q 2 ν + 4 ( ( ν + 3 ) 2 [ 5 ] [ 2 ν + 5 ] q − 2 q 4 ( ν + 5 ) [ 3 ] q [ 2 ν + 3 ] q ) .
The radius of starlikeness r ∗ ( g ν ( 2 ) ( z ; q 2 ) ) satisfies the inequality
1 6 q 2 ν + 4 [ 3 ] q [ 2 ν + 3 ] q < ( r ∗ ( g ν ( 2 ) ( z ; q 2 ) ) ) 2 < 3 [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q q 2 ν + 4 ( 9 [ 5 ] [ 2 ν + 5 ] q − 10 q 4 [ 3 ] q [ 2 ν + 3 ] q ) .
The radius of starlikeness r ∗ ( ζ ν ( 2 ) ( z ; q 2 ) ) satisfies the inequality
1 2 q 2 ν + 4 [ 3 ] q [ 2 ν + 3 ] q < r ∗ ( ζ ν ( 2 ) ( z ; q 2 ) ) < [ 5 ] q [ 3 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q q 2 ν + 4 ( 2 [ 5 ] q [ 2 ν + 5 ] q − 3 q 4 [ 3 ] q [ 2 ν + 3 ] q ) .

Corollary 3.5

Let ∣ ν ∣ < 1 2 . Then the following assertions hold

If 0 < q < 6 11 , then
r ∗ ( f ν ( 2 ) ( z ; q 2 ) ) ∈ q − ν − 2 2 7 [ 3 ] q [ 2 ν + 3 ] q , q − ν − 2 6 [ 5 ] q [ 2 ν + 5 ] q .
If 0 < q < 9 10 , then
r ∗ ( g ν ( 2 ) ( z ; q 2 ) ) ∈ q − ν − 2 1 6 [ 3 ] q [ 2 ν + 3 ] q , q − ν − 2 3 [ 5 ] q [ 2 ν + 5 ] q 9 − 10 q 2 .
If 0 < q < 2 3 , then
r ∗ ( ζ ν ( 2 ) ( z ; q 2 ) ) ∈ 1 2 q 2 ν + 4 [ 3 ] q [ 2 ν + 3 ] q , [ 5 ] q [ 2 ν + 5 ] q q 2 ν + 4 ( 2 − 3 q 2 ) .

4 Radii of convexity of the q-Bessel-Struve functions

The following result is essential for the proof of Theorem 7.

Lemma 4.1

[17] Let f be as in (2.1). If

∑ n = 2 ∞ n ( n − α ) ∣ a n ∣ ≤ 1 − α ,

then the function f is in the class K .

Theorem 7

Let ∣ ν ∣ < 1 2 , q ∈ ( 0 , 1 ) , α ∈ [ 0 , 1 ) . The following statements hold

If the inequality
(4.1) ( 2 − 2 ) q < 1 − q 2 ν + 1 1 − q < ( 2 + 2 ) q
holds, then the function ζ ν ( 3 ) ( z ; q 2 ) is convex of order α .
If the inequality
(4.2) ( 2 − 2 ) q 2 ν + 1 < 1 − q 2 ν + 1 1 − q < ( 2 + 2 ) q 2 ν + 1
holds, then the function ζ ν ( 3 ) ( z ; q 2 ) is convex of order α .

Proof

By the relation (2.3) and Lemma 4.1, it is enough to show that the following inequality
(4.3) k 2 = ∑ n ≥ 2 n ( n − α ) ( − 1 ) n − 1 q n ( n − 1 ) ( 1 − q ) 2 n − 1 ( 1 − q 2 ν + 1 ) ( q ; q 2 ) n ( q 2 ν + 1 ; q 2 ) n ≤ 1 − α
holds true under the hypothesis. By the same way in the proof of Theorem 1, we obtain
k 2 ≤ ∑ n ≥ 2 n ( n − α ) q n − 1 ( 1 − q ) n − 1 ( 1 − q 2 ν + 1 ) n − 1 = ∑ n ≥ 2 n 2 q ( 1 − q ) 1 − q 2 ν + 1 n − 1 − α ∑ n ≥ 2 n q ( 1 − q ) 1 − q 2 ν + 1 n − 1 .
From (4.1), we have
q ( 1 − q ) 1 − q 2 ν + 1 < 1 .
Therefore, by using (3.5) and the series sum
∑ n ≥ 2 n 2 r n − 1 = r 3 − 3 r 2 + 4 r ( 1 − r ) 3 ,
we obtain
k 2 ≤ q 3 ( 1 − q ) 3 − 3 q 2 ( 1 − q ) 2 ( 1 − q 2 ν + 1 ) + 4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 [ 1 − q 2 ν + 1 − q ( 1 − q ) ] 3 − α 2 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − q 2 ( 1 − q ) 2 [ 1 − q 2 ν + 1 − q ( 1 − q ) ] 2 .
To obtain k 2 ≤ 1 − α , we solve the inequality
q 3 ( 1 − q ) 3 − 3 q 2 ( 1 − q ) 2 ( 1 − q 2 ν + 1 ) + 4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 [ 1 − q 2 ν + 1 − q ( 1 − q ) ] 3 − α 2 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − q 2 ( 1 − q ) 2 [ 1 − q 2 ν + 1 − q ( 1 − q ) ] 2 ≤ 1 − α .
It implies that
2 q 3 ( 1 − q ) 3 − 6 q 2 ( 1 − q ) 2 ( 1 − q 2 ν + 1 ) + 7 q ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 − ( 1 − q 2 ν + 1 ) 3 1 − q 2 ν + 1 − q ( 1 − q ) ≤ α ( 4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − 2 q 2 ( 1 − q ) 2 − ( 1 − q 2 ν + 1 ) 2 ) .
Then
α ≤ 2 q 3 ( 1 − q ) 3 − 6 q 2 ( 1 − q ) 2 ( 1 − q 2 ν + 1 ) + 7 q ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 − ( 1 − q 2 ν + 1 ) 3 2 q 3 ( 1 − q ) 3 − 6 q 2 ( 1 − q ) 2 ( 1 − q 2 ν + 1 ) + 5 q ( 1 − q ) ( 1 − q 2 ν + 1 ) 2 − ( 1 − q 2 ν + 1 ) 3 < 1 ,
since for ∣ ν ∣ < 1 2 , we have
4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − 2 q 2 ( 1 − q ) 2 − ( 1 − q 2 ν + 1 ) 2 < 0 ,
as shown in Figure 1. That is,
4 q < 2 q 2 t + t ,
where t = 1 − q 2 ν + 1 1 − q . Hence,
( 2 − 2 ) q < 1 − q 2 ν + 1 1 − q < ( 2 + 2 ) q .
Therefore, the inequality (4.1) implies that k 2 ≤ 1 − α and so the function ζ ν ( 3 ) ( z ; q 2 ) is convex of order α . Similarly, we can prove part (b).□

$Figure 1 The graph of the function 4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − 2 q 2 ( 1 − q ) 2 − ( 1 − q 2 ν + 1 ) 2 4q\left(1-q)\left(1-{q}^{2\nu +1})-2{q}^{2}{\left(1-q)}^{2}-{\left(1-{q}^{2\nu +1})}^{2} , 0 < q < 1 0\lt q\lt 1 , ∣ ν ∣ < 1 ∕ 2 | \nu | \lt 1/2 .$

Figure 1

The graph of the function 4 q ( 1 − q ) ( 1 − q 2 ν + 1 ) − 2 q 2 ( 1 − q ) 2 − ( 1 − q 2 ν + 1 ) 2 , 0 < q < 1 , ∣ ν ∣ < 1 ∕ 2 .

Remark 4.2

In [Theorem 1, (b), 34], the author proved the following result for the convexity of the normalized second Jackson q-Bessel function h ν ( 2 ) ( z ; q ) ≔ 2 ν ( q ; q ) ∞ ( q ν + 1 ; q ) ∞ J ν ( 2 ) ( z ; q ) .

Let α ∈ [ 0 , 1 ) , ν > − 1 , and q ∈ ( 0 , 1 ) satisfy the condition 4 ( 1 − q ) ( 1 − q ν ) − q ν > 0 . Then if α satisfies the condition

(4.4) α ≥ 2 q 3 ν − 24 q 2 ν ( 1 − q ) ( 1 − q ν ) + 112 q ν ( 1 − q ) 2 ( 1 − q ν ) 2 − 64 ( 1 − q ) 3 ( 1 − q ν ) 3 2 q 3 ν − 24 q 2 ν ( 1 − q ) ( 1 − q ν ) + 80 q ν ( 1 − q ) 2 ( 1 − q ν ) 2 − 64 ( 1 − q ) 3 ( 1 − q ν ) 3 ,

then the function h ν ( 2 ) ( z ; q ) is convex of order α .

The result is obviously wrong because the condition (4.4) gives that α > 1 , which contradicts α ∈ [ 0 , 1 ) . The reason for this contradiction is that the author did not notice that the denominator in (4.4) is negative as shown in Figure 2. Therefore, he should reverse the order of the inequality in (4.4) to obtain

(4.5) α ≤ 2 q 3 ν − 24 q 2 ν ( 1 − q ) ( 1 − q ν ) + 112 q ν ( 1 − q ) 2 ( 1 − q ν ) 2 − 64 ( 1 − q ) 3 ( 1 − q ν ) 3 2 q 3 ν − 24 q 2 ν ( 1 − q ) ( 1 − q ν ) + 80 q ν ( 1 − q ) 2 ( 1 − q ν ) 2 − 64 ( 1 − q ) 3 ( 1 − q ν ) 3 .

Consequently, a corrected version for [Theorem 1, (b), 34] is

If ( 2 − 2 ) q ν < 4 ( 1 − q ) ( 1 − q ν ) < ( 2 + 2 ) q ν , then the normalized second Jackson q-Bessel function h ν ( 2 ) ( z ; q ) is convex of order α .

Similarly [Theorem 2, (b), 34] the statement is wrong as shown in Figure 3 and a corrected version is

If ( 2 − 2 ) q < ( 1 − q ) ( 1 − q ν ) < ( 2 + 2 ) q , then the normalized third Jackson q-Bessel function h ν ( 3 ) ( z ; q ) is convex of order α .

$Figure 2 The graph of the function 2 q 3 ν − 24 q 2 ν ( 1 − q ) ( 1 − q ν ) + 80 q ν ( 1 − q ) 2 ( 1 − q ν ) 2 − 64 ( 1 − q ) 3 ( 1 − q ν ) 3 2{q}^{3\nu }-24{q}^{2\nu }\left(1-q)\left(1-{q}^{\nu })+80{q}^{\nu }{\left(1-q)}^{2}{\left(1-{q}^{\nu })}^{2}-64{\left(1-q)}^{3}{\left(1-{q}^{\nu })}^{3} , 0 < q < 1 0\lt q\lt 1 , ν > 0 \nu \gt 0 .$

Figure 2

The graph of the function 2 q 3 ν − 24 q 2 ν ( 1 − q ) ( 1 − q ν ) + 80 q ν ( 1 − q ) 2 ( 1 − q ν ) 2 − 64 ( 1 − q ) 3 ( 1 − q ν ) 3 , 0 < q < 1 , ν > 0 .

$Figure 3 The graph of the function 2 q ( 1 − q ) ( 1 − q ν ) − q − ( ( 1 − q ) ( 1 − q ν ) − q ) 2 2\sqrt{q}\left(1-q)\left(1-{q}^{\nu })-q-{\left(\left(1-q)\left(1-{q}^{\nu })-\sqrt{q})}^{2} , 0 < q < 1 0\lt q\lt 1 , ν > 0 \nu \gt 0 .$

Figure 3

The graph of the function 2 q ( 1 − q ) ( 1 − q ν ) − q − ( ( 1 − q ) ( 1 − q ν ) − q ) 2 , 0 < q < 1 , ν > 0 .

Theorem 8

Let ∣ ν ∣ < 1 2 and α ∈ [ 0 , 1 ) . Then the following statements hold.

The radius of convexity of order α of f ν ( 3 ) ( z ; q 2 ) is the smallest positive root of the equation
z H ν ″ ( z ; q 2 ) H ν ′ ( z ; q 2 ) + 1 ν + 1 − 1 z H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) + 1 − α = 0 .
The radius of convexity of order α of g ν ( 3 ) ( z ; q 2 ) is the smallest positive root of the equation
z 2 H ν ″ ( z ; q 2 ) − ( 2 ν + α − 1 ) z H ν ′ ( z ; q 2 ) + ν ( ν + α ) H ν ( z ; q 2 ) = 0 .
The radius of convexity of order α of ζ ν ( 3 ) ( z ; q 2 ) is the smallest positive root of the equation
z H ν ″ ( z ; q 2 ) + ( 3 − 2 ( ν + α ) ) z H ν ′ ( z ; q 2 ) + ( 1 − ν ) ( 1 − ν − α ) H ν ( z ; q 2 ) = 0 .

Moreover, we have the inequalities

r α ( f ν ( 3 ) ) < h ν , 1 ′ < h ν , 1 , r α ( g ν ( 3 ) ) < β ν , 1 < h ν , 1 , r α ( ζ ν ( 3 ) ) < δ ν , 1 < h ν , 1 ,

where β ν , 1 and δ ν , 1 are the first positive zeros of the functions g ν ( 3 ) ′ ( z ; q 2 ) and ζ ν ( 3 ) ′ ( z ; q 2 ) , respectively.

Proof

Since
H ν ′ ( z ; q 2 ) = 1 1 + q ∑ n = 0 ∞ ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n + ν ,
then
( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − ν H ν ′ ( z ; q 2 ) = ν + 1 + ∑ n = 1 ∞ ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) Γ q 2 3 2 Γ q 2 ν + 3 2 Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n .
Hence,
1 ν + 1 ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 z − ν H ν ′ ( z ; q 2 ) = 1 + 1 ν + 1 ∑ n = 1 ∞ ( − 1 ) n q n 2 + n ( 2 n + ν + 1 ) Γ q 2 3 2 Γ q 2 ν + 3 2 Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 z 1 + q 2 n
is an entire function of order 1 2 .

On the other hand, (see [32]), if f is an entire function with non-integer infinite growth, then f has infinitely many zeros or f is a polynomial. Therefore, since the order of the real entire function z − ν H ν ′ ( z ; q 2 ) is 1 2 , then H ν ′ ( z ; q 2 ) has infinitely many zeros when ∣ ν ∣ < 1 2 . Now, by applying Hadamard’s theorem [6],
H ν ′ ( z ; q 2 ) = ( ν + 1 ) z ν ( 1 + q ) ν Γ q 2 1 2 Γ q 2 ν + 3 2 ∏ n ≥ 1 1 − z 2 h ν , n ′ 2 ,
where h ν , n ′ denotes the nth positive zero of H ν ′ ( z ; q 2 ) . It follows that
(4.6) 1 + z H ν ″ ( z ; q 2 ) H ν ′ ( z ; q 2 ) = 1 + ν − ∑ n ≥ 1 2 z 2 h ν , n ′ 2 − z 2 .
Consequently, by (3.9), we have
1 + z f ν ″ ( z ; q 2 ) f ν ′ ( z ; q 2 ) = 1 + z H ν ″ ( z ; q 2 ) H ν ′ ( z ; q 2 ) + 1 ν + 1 − 1 z H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) = 1 − 1 ν + 1 − 1 ∑ n ≥ 1 2 z 2 h ν , n 2 − z 2 − ∑ n ≥ 1 2 z 2 h ν , n ′ 2 − z 2 .
Now, suppose that ν ∈ − 1 2 , 0 . By using the inequality (3.10), for all z ∈ D h ν , 1 ′ ( q 2 ) , we obtain
(4.7) ℜ 1 + z f ν ″ ( z ; q 2 ) f ν ′ ( z ; q 2 ) ≥ 1 − 1 ν + 1 − 1 ∑ n ≥ 1 2 r 2 h ν , n 2 − r 2 − ∑ n ≥ 1 2 r 2 h ν , n ′ 2 − r 2 ,
where ∣ z ∣ = r . Moreover, if we use the inequality [15]
γ ℜ z a − z − ℜ z b − z ≥ γ ∣ z ∣ a − ∣ z ∣ − ∣ z ∣ a − ∣ z ∣ ,

a > b > 0 , γ ∈ [ 0 , 1 ] , and z ∈ C such that ∣ z ∣ < b ,
then we obtain inequality (4.7) when ν ∈ 0 , 1 2 . Here we used that the zeros h ν , n and h ν , n ′ interlace according to Theorem 2.2. The inequality (4.7) implies for r ∈ ( 0 , h ν , 1 ′ ( q 2 ) )
inf z ∈ D r ℜ 1 + z f ν ″ ( z ; q 2 ) f ν ′ ( z ; q 2 ) = 1 + r f ν ″ ( r ; q 2 ) f ν ′ ( r ; q 2 ) .
On the other hand, the function F ν ( ⋅ ; q 2 ) : ( 0 , h ν , 1 ′ ( q 2 ) ) → R , defined by
F ν ( r ; q 2 ) = 1 + r f ν ″ ( r ; q 2 ) f ν ′ ( r ; q 2 ) ,
is strictly decreasing for ∣ ν ∣ ≤ 1 2 and r < h ν , 1 ( q 2 ) h ν , 1 ′ ( q 2 ) . Since lim r → 0 + F ν ( r ; q 2 ) = 1 < α and lim r → h ν , 1 ′ ( q 2 ) − F ν ( r ; q 2 ) = − ∞ , then for z ∈ D r 1 , we have
ℜ 1 + z f ν ″ ( z ; q 2 ) f ν ′ ( z ; q 2 ) > α
if and only if r 1 is the smallest positive root of the equation
z H ν ″ ( z ; q 2 ) H ν ′ ( z ; q 2 ) + 1 ν + 1 − 1 z H ν ′ ( z ; q 2 ) H ν ( z ; q 2 ) + 1 − α = 0 .
Similarly, we can prove parts (b) and (c).□

Theorem 9

Let ∣ ν ∣ < 1 2 and α ∈ [ 0 , 1 ) . Then the following statements hold.

The radius of convexity of order α of f ν ( 2 ) ( z ; q 2 ) is the smallest positive root of the equation
z H ν ( 2 ) ″ ( z ∣ q 2 ) H ν ( 2 ) ′ ( z ∣ q 2 ) + 1 ν + 1 − 1 z H ν ( 2 ) ′ ( z ∣ q 2 ) H ν ( 2 ) ( z ∣ q 2 ) + 1 − α = 0 .
The radius of convexity of order α of g ν ( 2 ) ( z ; q 2 ) is the smallest positive root of the equation
z 2 H ν ( 2 ) ″ ( z ∣ q 2 ) − ( 2 ν + α − 1 ) z H ν ( 2 ) ′ ( z ∣ q 2 ) + ν ( ν + α ) H ν ( 2 ) ( z ∣ q 2 ) = 0 .
The radius of convexity of order α of ζ ν ( 2 ) ( z ; q 2 ) is the smallest positive root of the equation
z H ν ( 2 ) ″ ( z ; q 2 ) + ( 3 − 2 ( ν + α ) ) z H ν ( 2 ) ′ ( z ∣ q 2 ) + ( 1 − ν ) ( 1 − ν − α ) H ν ( 2 ) ( z ∣ q 2 ) = 0 .

Moreover, we have the inequalities

r α c ( f ν ( 2 ) ) < h ˜ ν , 1 ′ < h ˜ ν , 1 , r α c ( g ν ( 2 ) ) < β ˜ ν , 1 < h ˜ ν , 1 , r α c ( ζ ν ( 2 ) ) < δ ˜ ν , 1 < h ˜ ν , 1 ,

where β ˜ ν , 1 and δ ˜ ν , 1 are the first positive zeros of the functions g ν ( 2 ) ′ ( z ; q 2 ) and ζ ν ( 2 ) ′ ( z ; q 2 ) , respectively.

The following theorem gives the upper and the lower bound for the radii of convexity of the functions g ν ( 3 ) ( z ; q 2 ) and ζ ν ( 3 ) ( z ; q 2 ) .

Theorem 10

Then the following statements hold.

The radius of convexity r c ( g ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
1 3 q [ 3 ] q [ 2 ν + 3 ] q < r c ( g ν ( 3 ) ( z ; q 2 ) ) < 3 q [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q 81 [ 5 ] q [ 2 ν + 5 ] q − 50 q 2 [ 3 ] q [ 2 ν + 3 ] q .
The radius of convexity r c ( ζ ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
1 4 q 2 [ 3 ] q [ 2 ν + 3 ] q < r c ( ζ ν ( 3 ) ( z ; q 2 ) ) < 4 [ 3 ] q [ 2 ν + 3 ] q [ 5 ] q [ 2 ν + 5 ] q 16 q 2 [ 5 ] q [ 2 ν + 5 ] q − 18 q 4 [ 3 ] q [ 2 ν + 3 ] q .

Proof

By using the representation series of the function g ν ( 3 ) ( z ; q 2 ) , we obtain
Φ ν ( z ; q 2 ) ≔ ( z g ν ( 3 ) ′ ( z ; q 2 ) ) ′ = 1 + Γ q 2 3 2 Γ q 2 ν + 3 2 ∑ k ≥ 1 ( − 1 ) k q k 2 + k ( 2 k + 1 ) 2 Γ q 2 ( k + 3 2 ) Γ q 2 ( k + ν + 3 2 ) z 1 + q 2 k .
Since the function g ν ( 3 ) ( z ; q 2 ) belongs to the class ℒP of real entire functions, then Φ ν ( z ; q 2 ) also belongs to the class ℒP . Hence, the function Φ ν ( z ; q 2 ) has only real zeros. Also, its growth ρ is 0. Now, by applying Hadamard’s theorem, we obtain
Φ ν ( z ; q 2 ) = ∏ n ≥ 1 1 − z 2 β ¯ ν , n 2 ( q 2 ) ,
where β ¯ ν , n ( q 2 ) is the n th zero of the function Φ ν ( z ; q 2 ) . Then, we have
(4.8) Φ ν ′ ( z ; q 2 ) Φ ν ( z ; q 2 ) = − 2 ∑ k ≥ 0 a k + 1 z 2 k + 1 , ∣ z ∣ < β ¯ ν , 1 ( q 2 ) ,
where a k = ∑ n ≥ 1 ( β ¯ ν , n ( q 2 ) ) − 2 k . By using the infinite series of Φ ν ( z ; q 2 ) , we obtain
(4.9) Φ ν ′ ( z ; q 2 ) Φ ν ( z ; q 2 ) = ∑ n ≥ 0 A n z 2 n + 1 ∑ n ≥ 0 B n z 2 n ,
where
A n = ( − 1 ) n + 1 q n 2 + 3 n + 2 ( 2 n + 3 ) 2 ( 2 n + 2 ) Γ q 2 n + 5 2 Γ q 2 n + ν + 5 2 ( 1 + q ) 2 n + 2
and
B n = ( − 1 ) n q n 2 + n ( 2 n + 1 ) 2 Γ q 2 n + 3 2 Γ q 2 n + ν + 3 2 ( 1 + q ) 2 n .
By comparing the coefficient of (4.8) and (4.9), we have
A 0 = − 2 a 1 B 0 , A 1 = − 2 a 2 B 0 − 2 a 1 B 1 ,
which gives the following Rayleigh sums
a 1 = 9 q 2 [ 3 ] q [ 2 ν + 3 ] q
and
a 2 = q 4 ( 1 − q ) 4 [ 81 ( 1 − q 5 ) ( 1 − q 2 ν + 5 ) − 50 q 2 ( 1 − q 3 ) ( 1 − q 2 ν + 3 ) ] ( 1 − q 3 ) 2 ( 1 − q 5 ) ( 1 − q 2 ν + 3 ) 2 ( 1 − q 2 ν + 5 ) .
By using the Euler-Rayleigh inequality
a k − 1 k < β ¯ ν , 1 2 ( q 2 ) < a k a k + 1 ,
for k = 1 , we obtain
1 9 q 2 [ 3 ] q [ 2 ν + 3 ] q < ( r c ( g ν ( 3 ) ( z ; q 2 ) ) ) 2 < 9 [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q 81 q 2 [ 5 ] q [ 2 ν + 5 ] q − 50 q 4 [ 3 ] q [ 2 ν + 3 ] q .
Similarly, we can prove part (b).□

Corollary 4.3

Let ∣ ν ∣ < 1 2 . Then the following assertions hold

The radius of convexity r c ( g ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
1 3 q [ 3 ] q [ 2 ν + 3 ] q < r c ( g ν ( 3 ) ( z ; q 2 ) ) < 3 q [ 5 ] q [ 2 ν + 5 ] q 50 ( 1 − q 2 ) .
If 0 < q < 16 / 18 , then the radius of convexity r c ( ζ ν ( 3 ) ( z ; q 2 ) ) satisfies the inequality
1 4 q 2 [ 3 ] q [ 2 ν + 3 ] q < r c ( ζ ν ( 3 ) ( z ; q 2 ) ) < [ 5 ] q [ 2 ν + 5 ] q 4 q 2 1 − 18 16 q 2 .

Theorem 11

Then the following statements hold

The radius of convexity r c ( g ν ( 2 ) ( z ; q 2 ) ) satisfies the inequality
1 3 q − ν − 2 [ 3 ] q [ 2 ν + 3 ] q < r c ( g ν ( 2 ) ( z ; q 2 ) ) < 3 q − ν − 2 [ 3 ] q [ 5 ] q [ 2 ν + 3 ] q [ 2 ν + 5 ] q 81 [ 5 ] q [ 2 ν + 5 ] q − 50 q 4 [ 3 ] q [ 2 ν + 3 ] q .
The radius of convexity r c ( ζ ν ( 2 ) ( z ; q 2 ) ) satisfies the inequality
1 4 q 2 ν + 4 [ 3 ] q [ 2 ν + 3 ] q < r c ( ζ ν ( 2 ) ( z ; q 2 ) ) < 2 [ 3 ] q [ 2 ν + 3 ] q [ 5 ] q [ 2 ν + 5 ] q q 2 ν + 4 ( 8 [ 5 ] q [ 2 ν + 5 ] q − 9 q 4 [ 3 ] q [ 2 ν + 3 ] q ) .

Corollary 4.4

The following statements hold.

The radius of convexity r c ( g ν ( 2 ) ( z ; q 2 ) ) satisfies the inequality
1 3 q ν + 2 [ 3 ] q [ 2 ν + 3 ] q < r c ( g ν ( 2 ) ( z ; q 2 ) ) < 3 q ν + 2 [ 5 ] q [ 2 ν + 5 ] q 50 ( 1 − q 4 ) .
If 0 < q < 8 / 9 , then
1 4 q 2 ν + 4 [ 3 ] q [ 2 ν + 3 ] q < r c ( ζ ν ( 2 ) ( z ; q 2 ) ) < [ 5 ] q [ 2 ν + 5 ] q 4 q 2 ν + 4 ( 1 − 9 8 q 4 ) .

5 Conclusion

This paper investigates the starlikeness and convexity properties for three q-analogs of the normalized Bessel-Struve functions. We used their Hadamard factorization and applied Euler-Rayleigh inequality for the first zeros of these functions and their derivatives. As a result, we gave lower and upper bounds for the radii of starlikeness and convexity of these functions.

Acknowledgements

This project was supported financially by the Academy of Scientific Research and Technology (ASRT), Egypt, Grant No. 6595, and ASRT is the second affiliation of this research.

Author contributions: All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.
Conflict of interest: The authors declare that they have no competing interests.

References

[1] A. R. Ahmadi and S. E. Widnall, Unsteady lifting-line theory as a singular perturbation problem, J. Fluid Mech. 153 (1985), 59–81. 10.1017/S0022112085001148Search in Google Scholar

[2] D. C. Shaw, Perturbational results for diffraction of water-waves by nearly vertical barriers, J. Appl. Math. 34 (1985), 99–117. 10.1093/imamat/34.1.99Search in Google Scholar

[3] K. M. Oraby and Z. S. I. Mansour, On q-analogs of Struve functions, Quaest. Math. 44 (2021), no. 9, 1–29. 10.2989/16073606.2021.2011798Search in Google Scholar

[4] G. Gasper and M. Rahman, Basic Hypergeometric Series, Cambridge University Press, Cambridge, 2004. 10.1017/CBO9780511526251Search in Google Scholar

[5] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1944. Search in Google Scholar

[6] B. Ya. Levin, Lectures on Entire Functions, vol. 150, Translations of Mathematical Monographs, American Mathematical Society, 1996. 10.1090/mmono/150Search in Google Scholar

[7] I. Aktas, A. Baricz, and H. Orhan, Bounds for radii starlikeness and convexity of some special functions, Turk. J. Math. 42 (2018), 211–226. 10.3906/mat-1610-41Search in Google Scholar

[8] A. Baricz and R. Szász, Close-to-convexity of some special functions and their derivatives, Bull. Malays. Math. Sci. Soc. 39 (2016), 427–437. 10.1007/s40840-015-0180-7Search in Google Scholar

[9] A. Baricz and N. Yağmur, Geometric properties of some Lommel and Struve functions, Ramanujan J. 42 (2017), 325–346. 10.1007/s11139-015-9724-6Search in Google Scholar

[10] N. Yağmur and H. Orhan, Starlikeness and convexity of generalized Struve functions, Abstr. Appl. Anal. 2013 (2013), 1–7. 10.1155/2013/954513Search in Google Scholar

[11] I. Aktas, A. Baricz, and N. Yağmur, Bounds for the radii of univalence of some special functions, Math. Inequal. Appl. 20 (2017), 825–843. 10.7153/mia-20-52Search in Google Scholar

[12] A. Baricz, Geometric properties of generalized Bessel functions of complex order, Mathematica 48 (2006), 13–18. Search in Google Scholar

[13] A. Baricz, Geometric properties of generalized Bessel functions, Publ. Math. Debrecen 73 (2008), 155–178. 10.1007/978-3-642-12230-9_2Search in Google Scholar

[14] A. Baricz, P. A. Kupán, and R. Szász, The radius of starlikeness of normalized Bessel functions of the first kind, Proc. Amer. Math. Soc. 142 (2014), no. 2019–2025, 2014. 10.1090/S0002-9939-2014-11902-2Search in Google Scholar

[15] A. Baricz and R. Szász, The radius of convexity of normalized Bessel functions of the first kind, Anal. Appl. 12 (2014), no. 5, 485–509. 10.1142/S0219530514500316Search in Google Scholar

[16] A. Baricz, D. K. Dimitrov, H. Orhan, and N. Yağmur, Radii of starlikeness of some special functions, Proc. Amer. Math. Soc. 144 (2014), no. 8, 3355–3367. 10.1090/proc/13120Search in Google Scholar

[17] H. Silverman, Starlike and convexity properties for hypergeometric functions, J. Math. Anal. Appl. 172 (1993), 574–581. 10.1006/jmaa.1993.1044Search in Google Scholar

[18] E. Deniz, S. Kazimoğlu, and M. Çağlar, Radii of uniform convexity of Lommel and Struve functions, Bull. Iran. Math. Soc. 47 (2021), 1533–1557. 10.1007/s41980-020-00457-8Search in Google Scholar

[19] M. Çağlar, E. Deniz, and R. Szász, Radii of α-convexity of some normalized Bessel functions of the first kind, Results Math. 72 (2017), 12023–12035. 10.1007/s00025-017-0738-9Search in Google Scholar

[20] E. Deniz, A. Baricz, and M. Çağlar, Radii of convexity of some normalized Bessel functions of the first kind, Bull. Malays. Math. Sci. Soc. 83 (2015), 1255–1280. 10.1007/s40840-014-0079-8Search in Google Scholar

[21] M. E. H. Ismail, E. Merkes, and D. Styer, A generalization of starlike functions, Complex Var. Theory Appl. 14 (1990), 77–88. 10.1080/17476939008814407Search in Google Scholar

[22] I. Aktas and H. Orhan, Bounds for radii of convexity of some q-Bessel functions, Bull. Korean Math. Soc. 57 (2020), 355–369. Search in Google Scholar

[23] A. Baricz, D. K. Dimitrov, and I. Mecő, Radii of starlikeness and convexity of some q-Bessel functions, J. Math. Anal. Appl. 435 (2016), 968–985. 10.1016/j.jmaa.2015.10.065Search in Google Scholar

[24] E. Toklu, Radii of starlikeness and convexity of q-Mittage-Leffler functions, Turkish J. Math. 43 (2019), 2610–2630. 10.3906/mat-1907-54Search in Google Scholar

[25] E. Toklu, I. Aktas, and H. Orhan, Radii problems for normalized q-Bessel and Wright functions, Acta Univ. Sapientiae Math. 11 (2019), 203–223. 10.2478/ausm-2019-0016Search in Google Scholar

[26] A. Baricz, S. Ponnusamy, and S. Singh, Turán type inequalities for Struve functions, J. Math. Anal. App. 455 (2017), 971–984. 10.1016/j.jmaa.2016.08.026Search in Google Scholar

[27] P. I. Duren, Univalent Functions, Springer-Verlag, New York, 1983. Search in Google Scholar

[28] S. Agarwal and S. K. Sahoo, Generalization of starlike functions of order alpha, Hokkaido Math. J. 46 (2017), 15–27. 10.14492/hokmj/1498788094Search in Google Scholar

[29] D. K. Dimitrov and Y. B. Cheikh, Laguerre polynomials as Jensen polynomials of Laguerre-Pólya entire functions, J. Comput. Appl. Math. 233 (2009), no. 3, 703–707. 10.1016/j.cam.2009.02.039Search in Google Scholar

[30] H. Skovgarrd, On inequalities of the Turán type, Math. Scand. 2 (1954), 65–73. 10.7146/math.scand.a-10396Search in Google Scholar

[31] B. Dahlberg, A minimum principle for positive harmonic function, Proc. London Math. Soc. 3 (1976), no. 33, 238–250. 10.1112/plms/s3-33.2.238Search in Google Scholar

[32] R. P. Boas Jr., Entire Functions, Academic Press Inc., New York, 1954. Search in Google Scholar

[33] M. E. H. Ismail and M. E. Muldoon, Bounds for the small real and purely imaginary zeros of Bessel and related functions, Methods Appl. Anal. 2 (1995), 1–21. 10.4310/MAA.1995.v2.n1.a1Search in Google Scholar

[34] I. Aktas, On some geometric properties and Hardy class of q-Bessel functions, AIMS Math. 4 (2020), 3156–3168. 10.3934/math.2020203Search in Google Scholar

Received: 2021-08-16

Accepted: 2022-02-14

Published Online: 2022-04-19

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2022-0004

Keywords for this article

starlike; convexity; q-Bessel-Struve functions

Creative Commons

BY 4.0