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Article

# Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

by
Shahid Mahmood
1,
Hari M. Srivastava
2,3,
Nazar Khan
4,
4,*,
Bilal Khan
4 and
Irfan Ali
5
1
Department of Mechanical Engineering, Sarhad University of Science and Information Technology, Ring Road, Peshawar 25000, Pakistan
2
Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada
3
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad 22010, Pakistan
5
Department of Mathematical Sciences, Balochistan University of Information Technology, Engineering and Management Sciences, Quetta 87300, Pakistan
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(3), 347; https://doi.org/10.3390/sym11030347
Submission received: 1 January 2019 / Revised: 23 February 2019 / Accepted: 23 February 2019 / Published: 7 March 2019
(This article belongs to the Special Issue Integral Transforms and Operational Calculus)

## Abstract

:
The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.
MSC:
Primary 05A30, 30C45; Secondary 11B65, 47B38

## 1. Introduction

We denote by $A U$ the class of functions which are analytic in the open unit disk
$U = z : z ∈ C and z < 1 ,$
where $C$ is the complex plane. Let $A$ be the class of analytic functions having the following normalized form:
$f z = z + ∑ n = 2 ∞ a n z n ∀ z ∈ U$
in the open unit disk $U ,$ centered at the origin and normalized by the conditions given by
$f 0 = 0 and f ′ 0 = 1 .$
In addition, let $S ⊂ A$ be the class of functions which are univalent in $U$. The class of starlike functions in $U$ will be denoted by $S * ,$ which consists of normalized functions $f ∈ A$ that satisfy the following inequality:
$ℜ z f ′ z f z > 0 , ∀ z ∈ U .$
If two functions f and g are analytic in $U$, we say that the function f is subordinate to g and write in the form:
$f ≺ g or f z ≺ g z ,$
if there exists a Schwarz function w which is analytic in $U$, with
$w 0 = 0 and w z < 1 ,$
such that
$f z = g w z .$
In particular, if the function g is univalent in $U$, then it follows that (cf., e.g., [1]; see also [2])
$f ( z ) ≺ g ( z ) ( z ∈ U ) ⇒ f ( 0 ) = g ( 0 ) and f ( U ) ⊂ g ( U ) .$
Moreover, for two analytic functions f and g given byx
$f ( z ) = z + ∑ n = 2 ∞ a n z n ∀ z ∈ U$
and
$g ( z ) = z + ∑ n = 2 ∞ b n z n ∀ z ∈ U ,$
the convolution (or the Hadamard product) of f and g is defined as follows:
$f ( z ) ∗ g ( z ) = z + ∑ n = 2 ∞ a n b n z n .$
We next denote by $P$ the class of analytic functions p which are normalized by
$p z = 1 + ∑ n = 1 ∞ p n z n ,$
such that
$ℜ p z > 0 ( z ∈ U ) .$
We now recall some essential definitions and concept details of the basic or quantum (q-) calculus, which are used in this paper. We suppose throughout the paper that $0 < q < 1$ and that
$N = 1 , 2 , 3 , ⋯ = N 0 ∖ 0 ( N 0 = 0 , 1 , 2 , 3 , ⋯ ) .$
Definition 1.
Let $q ∈ 0 , 1$ and define the q-number $λ q$ by
$λ q = 1 − q λ 1 − q ( λ ∈ C ) ∑ k = 0 n − 1 q k = 1 + q + q 2 + ⋯ + q n − 1 ( λ = n ∈ N ) .$
Definition 2.
Let $q ∈ 0 , 1$ and define the q-factorial $n q !$ by
$n q ! = 1 ( n = 0 ) ∏ k = 1 n − 1 k q ( n ∈ N ) .$
Definition 3.
Let $q ∈ 0 , 1$ and define the generalized q-Pochhammer symbol $λ q , n$ by
$λ q , n = 1 ( n = 0 ) ∏ k = 0 n λ + k q ( n ∈ N ) .$
Definition 4.
For $ω > 0 ,$ let the q-gamma function $Γ q ( ω )$ be defined by
$Γ q ω + 1 = ω q Γ q ω and Γ q 1 : = 1 .$
Definition 5.
(see [3,4]) The q-derivative (or the q-difference) operator $D q$ of a function f in a given subset of $C$ is defined by
$D q f z = f z − f q z 1 − q z z ≠ 0 f ′ 0 z = 0 ,$
provided that $f ′ 0$ exists.
We note from Definition 5 that
$lim q → 1 − D q f z = lim q → 1 − f q z − f z 1 − q z = f ′ z ,$
for a differentiable function f in a given subset of $C$. It is readily deduced from (1) and (4) that
$D q f z = 1 + ∑ n = 2 ∞ n q a n z n − 1 .$
The operator $D q$ plays a vital role in the investigation and study of numerous subclasses of the class of analytic functions of the form given in Definition 5. A q-extension of the class of starlike functions was first introduced in [5] by using the q-derivative operator (see Definition 6 below). A background of the usage of the q-calculus in the context of Geometric Funciton Theory was actually provided and the basic (or q-) hypergeometric functions were first used in Geometric Function Theory by Srivastava (see, for details, [6]). Some recent investigations associated with the q-derivative operator $D q$ in analytic function theory can be found in [7,8,9,10,11,12,13] and the references cited therein.
Definition 6.
(see [5]) A function $f ∈ A U$ is said to belong to the class $S q ∗$ if
$f 0 = f ′ 0 − 1 = 0$
and
$z f z D q f z − 1 1 − q ≦ 1 1 − q ∀ z ∈ U .$
The notation $S q ∗$ was first used by Sahoo et al. (see [14]).
It is readily observed that, as $q → 1 −$, the closed disk given
$w − 1 1 − q ≦ 1 1 − q$
becomes the right-half plane and the class $S q ∗$ reduces to $S ∗ .$ Equivalently, by using the principle of subordination between analytic functions, we can rewrite the conditions in (6) and (7) as follows (see [15]):
$z f z D q f z ≺ p ^ p ^ = 1 + z 1 − q z .$
Definition 7.
(see [16]) For a function $f ∈ A U ,$ the Ruscheweyh-type q-derivative operator is defined as follows:
$R q δ f z = ϕ q , δ + 1 ; z ∗ f z = z + ∑ n = 2 ∞ ψ n − 1 a n z n z ∈ U ; δ > − 1 ,$
where
$ϕ q , δ + 1 ; z = z + ∑ n = 2 ∞ ψ n − 1 z n$
and
$ψ n − 1 = Γ q δ + n n − 1 q ! Γ q δ + 1 = n + 1 n − 1 , q n − 1 q ! .$
From (8) it can be seen that
$R q 0 f z = f z and R q 1 f z = z D q f z ,$
$R q m f z = z D q m f z z m − 1 f z m q ! ( m ∈ N ) ,$
$lim q → 1 − ϕ q , δ + 1 ; z = z 1 − z δ + 1$
and
$lim q → 1 − R q δ f z = f z ∗ z 1 − z δ + 1 .$
This shows that, in case of $q → 1 −$, the Ruscheweyh-type q-derivative operator reduces to the Ruscheweyh derivative operator $D δ f ( z )$ (see [17]). From (8) the following identity can easily be derived:
$z D q R q δ f z = 1 + δ q q δ R q δ + 1 f z − δ q q δ R q δ f z .$
If $q → 1 − ,$ then
$z R δ f z ′ = 1 + δ R δ + 1 f z − δ R δ f z .$
Now, by using the Ruscheweyh-type q-derivative operator, we define the following class of q-starlike functions.
Definition 8.
For $f ∈ A U ,$ we say that f belongs to the class $RS q * δ$ if the following inequality holds true:
$z D q R q δ f z f z − 1 1 − q ≦ 1 1 − q z ∈ U ; δ > − 1$
or, equivalently, we have (see [15])
$z D q R q δ f z f z ≺ 1 + z 1 − q z$
by using the principle of subordination.
Let $n ≧ 0$ and $j ≧ 1$. The $j th$ Hankel determinant is defined as follows:
$H j n = a n a n + 1 . . . a n + j − 1 a n + 1 . . . . . . . . . . . a n + j − 1 . . . . a n + 2 j − 1$
The above Hankel determinant has been studied by several authors. In particular, sharp upper bounds on $H 2 2$ were obtained by several authors (see, for example, [18,19,20,21]) for various classes of normalized analytic functions. It is well-known that the Fekete-Szegö functional $a 3 − a 2 2 = H 2 1$. This functional is further generalized as $a 3 − μ a 2 2$ for some real or complex $μ$. In fact, Fekete and Szegö gave sharp estimates of $a 3 − μ a 2 2$ for real $μ$ and $f ∈ S$, the class of normalized univalent functions in $U$. It is also known that the functional $a 2 a 4 − a 3 2$ is equivalent to $H 2 2$. Babalola [22] studied the Hankel determinant $H 3 1$ for some subclasses of analytic functions. In the present investigation, our focus is on the Hankel determinant $H 3 1$ for the above-defined function class $RS q * δ .$

## 2. A Set of Lemmas

Each of the following lemmas will be needed in our present investigation.
Lemma 1.
(see [23]) Let
$p ( z ) = 1 + c 1 z + c 2 z 2 + ⋯$
be in the class $P$ of functions with positive real part in $U$. Then, for any complex number $υ ,$
$c 2 − υ c 1 2 ≦ − 4 υ + 2 υ ≦ 0 2 0 ≦ υ ≦ 1 4 υ − 2 υ ≧ 1 .$
When $υ < 0$ or $υ > 1 ,$ the equality holds true in $( 13 )$ if and only if
$p ( z ) = 1 + z 1 − z$
or one of its rotations. If $0 < υ < 1 ,$ then the equality holds true in (13) if and only if
$p ( z ) = 1 + z 2 1 − z 2$
or one of its rotations. If $υ = 0 ,$ the equality holds true in (13) if and only if
$p ( z ) = 1 + ρ 2 1 + z 1 − z + 1 − ρ 2 1 − z 1 + z 0 ≦ ρ ≦ 1$
or one of its rotations. If $υ = 1 ,$ then the equality in (13) holds true if $p ( z )$ is a reciprocal of one of the functions such that the equality holds true in the case when $υ = 0 .$
Lemma 2.
(see [24,25]) Let
$p ( z ) = 1 + p 1 z + p 2 z 2 + ⋯$
be in the class $P$ of functions with positive real part in $U$. Then
$2 p 2 = p 1 2 + x 4 − p 1 2$
for some $x ,$ $x ≦ 1$ and
$4 p 3 = p 1 3 + 2 4 − p 1 2 p 1 x − 4 − p 1 2 p 1 x 2 + 2 4 − p 1 2 1 − x 2 z$
for some z $( z ≦ 1 ) .$
Lemma 3.
(see [26]) Let
$p ( z ) = 1 + p 1 z + p 2 z 2 + ⋯$
be in the class $P$ of functions positive real part in $U .$ Then
$p k ≦ 2 k ∈ N$
and the inequality is sharp.

## 3. Main Results

In this section, we will prove our main results. Throughout our discussion, we assume that
$q ∈ 0 , 1 and δ > − 1 .$
Our first main result is stated as follows.
Theorem 1.
Let $f ∈ R S q ∗ δ$ be of the form (1). Then
$a 3 − μ a 2 2 ≦ 1 + q + q 2 ψ 1 2 − μ 1 + q 2 ψ 2 q 2 ψ 2 ψ 1 2 μ < q 2 + 1 ψ 1 2 1 + q 2 ψ 2 1 q ψ 2 q 2 + 1 ψ 1 2 1 + q 2 ψ 2 ≦ μ ≦ ψ 1 2 ψ 2 μ 1 + q 2 ψ 2 − 1 + q + q 2 ψ 1 2 q 2 ψ 2 ψ 1 2 μ > ψ 1 2 ψ 2 ,$
where $ψ n − 1$ is given by (10).
It is also asserted that, for
$q 2 + 1 ψ 1 2 1 + q 2 ψ 2 ≦ μ ≦ 1 + q + q 2 ψ 1 2 1 + q 2 ψ 2 ,$
$| a 3 − μ a 2 2 | + μ − q 2 + 1 ψ 1 2 1 + q 2 ψ 2 | a 2 | 2 ≤ 1 q ψ 2$
and that, for
$1 + q + q 2 ψ 1 2 1 + q 2 ψ 2 ≦ μ ≦ ψ 1 2 ψ 2 ,$
$| a 3 − μ a 2 2 | + ψ 1 2 − μ ψ 2 ψ 2 | a 2 | 2 ≦ 1 q ψ 2 .$
Proof.
If $f ∈ RS q ∗ δ$, then it follows from (12) that
$z D q R q δ f z f z ≺ ϕ z ,$
where
$ϕ z = 1 + z 1 − q z .$
We define a function $p ( z )$ by
$p z = 1 + w z 1 − w z = 1 + p 1 z + p 2 z 2 + p 3 z 3 + ⋯ .$
It is clear that $p ∈ P$. From the above equation, we have
$w z = p z − 1 p z + 1 .$
From (14), we find that
$z D q R q δ f z f z = ϕ w z ,$
together with
$ϕ w z = 2 p z 1 − q p z + 1 + q .$
Now
$2 p z 1 − q p z + 1 + q = 1 + 1 2 ( 1 + q ) p 1 z + 1 2 ( q + 1 ) p 2 − 1 4 ( 1 − q 2 ) p 1 2 z 2 + 1 2 ( 1 + q ) p 3 − 1 2 ( 1 − q 2 ) p 1 p 2 + 1 8 ( 1 + q ) ( 1 − q ) 2 p 1 3 z 3 + { 1 2 1 + q p 4 = 1 4 1 − q 2 p 2 2 − 1 2 1 − q 2 p 1 p 3 + 3 8 ( 1 + q ) ( q − 1 ) 2 p 1 2 p 2 + 1 16 ( 1 + q ) ( 1 − q ) 3 p 1 4 } z 4 + ⋯ .$
Similarly, we get
$z D q R q δ f z R q δ f z = 1 + q a 2 ψ 1 z + q + q 2 ψ 2 a 3 − q ψ 1 2 a 2 2 z 2 + { q + q 2 + q 3 ψ 3 a 4 − 2 q + q 2 ψ 1 ψ 2 a 2 a 3 + q ψ 1 3 a 2 3 } z 3 + { q + q 2 + q 3 + q 4 ψ 5 a 5 − 2 q + q 2 + q 3 ψ 2 ψ 3 a 2 a 4 − q + q 2 ψ 2 2 a 3 2 + 3 q + q 2 ψ 1 2 ψ 2 a 2 2 a 3 − q ψ 1 4 a 2 4 } z 4 + ⋯ ,$
Therefore, we have
$a 2 = 1 + q 2 q ψ 1 p 1 ,$
$a 3 = 1 2 q ψ 2 p 2 + q 2 + 1 4 q 2 ψ 2 p 1 2$
and
$a 4 = 1 + q 2 q 1 + q + q 2 ψ 3 p 3 − 1 + q q − 2 2 q + 1 4 q 2 1 + q + q 2 ψ 3 p 1 p 2 + 1 + q q 2 + 1 q 2 − q + 1 8 q 3 1 + q + q 2 ψ 3 p 1 3 .$
We thus obtain
$a 3 − μ a 2 2 = 1 2 q ψ 2 p 2 − μ 1 + q 2 ψ 2 − 1 + q 2 ψ 1 2 2 q ψ 1 2 p 1 2 .$
Finally, by applying Lemma 1 and Equation (13) in conjunction with (18), we obtain the result asserted by Theorem 1. □
We now state and prove Theorem 2 below.
Theorem 2.
Let $f ∈ R S q ∗ δ$ be of the form (1). Then
$a 2 a 4 − a 3 2 ≦ 1 q 2 ψ 2 2 .$
Proof.
From (15)–(17), we obtain
$a 2 a 4 − a 3 2 = 1 + q 2 4 q 2 1 + q + q 2 ψ 1 ψ 3 p 1 p 3 − 1 + q 2 q − 2 2 q + 1 8 q 3 1 + q + q 2 ψ 1 ψ 3 + q 2 + 1 4 q 3 ψ 2 2 p 1 2 p − 1 4 q 2 ψ 2 2 p 2 2 + − q 2 + 1 2 16 q 4 ψ 2 2 + 1 + q 2 q 2 + 1 q 2 − q + 1 16 q 3 1 + q + q 2 ψ 1 ψ 3 p 1 4 .$
By using Lemma 2, we have
$a 2 a 4 − a 3 2 = 1 + q 2 q 2 + 1 q 2 − q + 1 16 q 3 1 + q + q 2 ψ 1 ψ 3 − q 2 + 1 2 16 q 4 ψ 2 2 p 1 4 + 1 + q 2 16 q 2 1 + q + q 2 ψ 1 ψ 3 p 1 p 1 3 + 2 p 1 4 − p 1 2 x − p 1 4 − p 1 2 x 2 + 2 4 − p 1 2 1 − x 2 z + q 2 + 1 8 q 3 ψ 2 2 · 1 + q 2 q − 2 2 q + 1 16 q 3 1 + q + q 2 ψ 1 ψ 3 p 1 2 p 1 2 + 4 − p 1 2 x − 1 16 q 2 ψ 2 2 p 1 4 + 4 − p 1 2 2 x 2 + 2 p 1 2 4 − p 1 2 x .$
Now, taking the moduli and replacing $x$ by $ρ$ and $p 1$ by $p ,$ we have
$a 2 a 4 − a 3 2 ≦ 1 Λ q ω q p 4 + 2 q 1 + q 2 ψ 2 2 p 4 − p 2 + Ω q 4 − p 2 p 2 ρ + q q + 1 2 ψ 2 2 p 2 + q 4 − p 2 · 1 + q + q 2 ψ 1 ψ 3 − 2 q 1 + q 2 ψ 2 2 p 4 − p 2 ρ 2 = F ( p , ρ ) ,$
where
$Λ q = 16 q 3 1 + q + q 2 ψ 1 ψ 3 ψ 2 2 ,$
$ω q = 3 + 3 q − q 3 + q 4 1 + q 2 ψ 2 2 − 1 + 3 q + 2 q 2 + 2 q 3 + q 4 · 1 + q + q 2 ψ 1 ψ 3$
and
$Ω q = 1 + q 2 2 q 2 − 5 q − 2 ψ 2 2 + 2 q q 2 + 2 1 + q + q 2 ψ 1 ψ 3 .$
Upon differentiating both sides $( 19 )$ with respect to $ρ$, we have
$∂ F ( p , ρ ) ∂ ρ = 1 Λ q Ω q 4 − p 2 p 2 + 2 q q + 1 2 ψ 2 2 p 2 + q 4 − p 2 · 1 + q + q 2 ψ 1 ψ 3 − 2 q 1 + q 2 ψ 2 2 p 4 − p 2 ρ .$
It is clear that
$∂ F ( p , ρ ) ∂ ρ > 0 ,$
which show that $F ( p , ρ )$ is an increasing function of $ρ$ on the closed interval $0 , 1 .$ This implies that the maximum value occurs at $ρ = 1 .$ This implies that
$max { F ( p , ρ ) } = F ( p , 1 ) = : G ( p ) .$
We now observe that
$G ( p ) = 1 Λ q ω q − Ω q − q q + 1 2 ψ 2 2 + q + q 2 + q 3 ψ 1 ψ 3 p 4 + 4 Ω q + 4 q q + 1 2 ψ 2 2 − 8 q + q 2 + q 3 ψ 1 ψ 3 p 2 + 16 q + q 2 + q 3 ψ 1 ψ 3 = G p .$
By differentiating both sides of $( 20 )$ with respect to $p ,$ we have
$G ′ ( p ) = 1 Λ q 4 ω q − Ω q − q q + 1 2 ψ 2 2 + q + q 2 + q 3 ψ 1 ψ 3 p 3 + 2 4 Ω q + 4 q q + 1 2 ψ 2 2 − 8 q + q 2 + q 3 ψ 1 ψ 3 p .$
Differentiating the above equation once again with respect to $p ,$ we get
$G ″ ( p ) = 1 Λ q 12 ω q − Ω q − q q + 1 2 ψ 2 2 + q + q 2 + q 3 ψ 1 ψ 3 p 2 + 2 4 Ω q + 4 q q + 1 2 ψ 2 2 − 8 q + q 2 + q 3 ψ 1 ψ 3 .$
For $p = 0 ,$ this shows that the maximum value of $( G ( p ) )$ occurs at $p = 0 .$ Hence, we obtain
$a 2 a 4 − a 3 2 ≦ 1 q 2 ψ 2 2 .$
The proof of Theorem 2 is thus completed. □
If, in Theorem 2, we let $q ⟶ 1 −$ and put $δ = 1 ,$ then we are led to the following known result.
Corollary 1.
(see [18]) Let $f ∈ S ∗$. Then
$a 2 a 4 − a 3 2 ≦ 1 ,$
and the inequality is sharp.
Theorem 3.
Let $f ∈ RS q ∗ δ$. Then
$a 2 a 3 − a 4 ≦ 1 + q κ q ψ 1 ψ 2 ψ 3 q 2 + q 3 + q 4 ,$
where
$κ q = 1 + q + q 2 2 ψ 3 − q 4 − 3 q + 6 q 2 + q + 1 ψ 1 ψ 2 .$
Proof.
Using the values given in $( 15 )$ and $( 16 )$ we have
$a 2 a 3 − a 4 = 1 + q q 2 + 1 8 q 3 ψ 1 ψ 2 − 1 + q q 2 + 1 q 2 − q + 1 8 ψ 3 q 2 + q 3 + q 4 p 1 3 + 1 + q 4 q 2 ψ 1 ψ 2 − q − 2 2 q + 1 1 + q 4 ψ 3 q 2 + q 3 + q 4 p 1 p 2 − 1 + q 2 q + q 2 + q 3 ψ 3 p 3 .$
We now use Lemma 2 and assume that $p 1 ≦ 2$. In addition, by Lemma 3, we let $p 1 = p$ and assume without restriction that $p ∈ 0 , 2 .$ Then, by taking the moduli and applying the trigonometric inequality on $( 22 )$ with $ρ = x ,$ we obtainx
$a 2 a 3 − a 4 ≦ 1 + q 8 q 3 + q 4 + q 5 ψ 1 ψ 2 ψ 3 κ q p 3 + η q p ( 4 − p 2 ) ρ + 2 q 2 ψ 1 ψ 2 ( 4 − p 2 ) + q 2 ψ 1 ψ 2 p − 2 ( 4 − p 2 ) ρ 2 = : F ( ρ ) ,$
where
$η q = q + q 2 + q 3 ψ 3 + 2 q 3 − q 2 − 2 q ψ 1 ψ 2$
and $κ q$ is given by (21). Differentiating $F ( ρ )$ with respect to $ρ$, we have
$F ′ ( ρ ) = 1 + q 8 q 3 + q 4 + q 5 ψ 1 ψ 2 ψ 3 η q p ( 4 − p 2 ) + 2 q 2 ψ 1 ψ 2 p − 2 ( 4 − p 2 ) ρ > 0 .$
This implies that $F ( ρ )$ is an increasing function of $ρ$ on the closed interval $0 , 1$. Hence, we have
$F ( ρ ) ≦ F ( 1 ) ( ∀ ρ ∈ 0 , 1 ) ,$
that is,
$F ( ρ ) ≦ 1 + q 8 q 3 + q 4 + q 5 ψ 1 ψ 2 ψ 3 κ q − η q − q 2 ψ 1 ψ 2 p 3 + 4 η q + 4 q 2 ψ 1 ψ 2 p = : G ( p ) .$
Since $p ∈ 0 , 2 ,$ $p = 2$ is a point of maximum. We thus obtain
$G p ≦ 1 + q κ q q 3 + q 4 + q 5 ψ 1 ψ 2 ψ 3 ,$
which corresponds to $ρ = 1$ and $p = 2$ and it is the desired upper bound. □
For $δ = 1$ and $q → 1 −$, we obtain the following special case of Theorem 3.
Corollary 2.
(see [22]) Let $f ∈ S ∗ .$ Then
$a 2 a 3 − a 4 ≦ 2 .$
Finally, we prove Theorem 4 below.
Theorem 4.
Let $f ∈ RS q ∗ δ$. Then
$H 3 ( 1 ) ≦ 1 + q + q 2 q 4 ψ 2 3 + ϰ q κ q q 5 1 + q + q 2 2 ψ 1 ψ 2 ψ 3 2 + τ q q 5 1 + q + q 2 + q 3 1 + q + q 2 ψ 2 ψ 4 ,$
where
$ϰ q = 1 + q 2 q 4 − 3 q 3 + 6 q 2 + q + 1 ,$
$τ q = 1 + q 4 q 7 + 2 q 6 + 6 q 5 + 7 q 4 + 13 q 3 − q − 1$
and $κ q$ is given by (21).
Proof.
Since
$H 3 ( 1 ) ≦ a 3 a 2 a 4 − a 3 2 + a 4 a 2 a 3 − a 4 + a 5 a 3 − a 2 2 ,$
by using Lemma 3, we have
$a 4 ≦ 1 + q 1 + q + 6 q 2 − 3 q 3 + q 4 q 3 1 + q + q 2 ψ 3$
and
$a 5 ≦ τ q q 4 1 + q + q 2 + q 3 1 + q + q 2 ψ 4 ,$
where $τ q$ is given by $( 24 )$. Now, by applying Theorems 1–3, we have the required result asserted by Theorem 4. □

## 4. Conclusions

By making use of the basic or quantum (q-) calculus, we have introduced a Ruscheweyh-type q-derivative operator. This Ruscheweyh-type q-derivative operator is then applied to define a certain subclass of q-starlike functions in the open unit disk $U$. We have successfully derived the upper bound of the third Hankel determinant for this family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. Our main results are stated and proved as Theorems 1–4. These general results are motivated essentially by their several special cases and consequences, some of which are pointed out in this presentation.

## Author Contributions

All authors contributed equally to the present investigation.

## Funding

This work is partially supported by Sarhad University of Science and I.T, Ring Road, Peshawar 2500, Pakistan.

## Acknowledgments

The first author would like to acknowledge Salim ur Rehman, V.C. Sarhad University of Science & I. T, for providing excellent research and academic environment.

## Conflicts of Interest

The authors declare no conflict of interest.

## References

1. Miller, S.S.; Mocanu, P.T. Differential subordination and univalent functions. Mich. Math. J. 1981, 28, 157–171. [Google Scholar] [CrossRef]
2. Miller, S.S.; Mocanu, P.T. Mocanu. In Differential Subordination: Theory and Applications; Series on Monographs and Textbooks in Pure and Applied Mathematics, No. 225; Marcel Dekker Incorporated: New York, NY, USA; Basel, Switzerland, 2000. [Google Scholar]
3. Jackson, F.H. On q-definite integrals. Quart. J. Pure Appl. Math. 1910, 41, 193–203. [Google Scholar]
4. Jackson, F.H. q-difference equations. Am. J. Math. 1910, 32, 305–314. [Google Scholar] [CrossRef]
5. Ismail, M.E.H.; Merkes, E.; Styer, D. A generalization of starlike functions. Complex Var. Theory Appl. 1990, 14, 77–84. [Google Scholar] [CrossRef]
6. Srivastava, H.M. Univalent functions, fractional calculus, and associated generalized hypergeometric functions. In Univalent Functions, Fractional Calculus, and Their Applications; Srivastava, H.M., Owa, S., Eds.; Halsted Press (Ellis Horwood Limited, Chichester); John Wiley and Sons: New York, NY, USA; Chichester, UK; Brisbane, Australian; Toronto, ON, Canada, 1989; pp. 329–354. [Google Scholar]
7. Srivastava, H.M.; Ahmad, Q.Z.; Khan, N.; Khan, N.; Khan, B. Hankel and Toeplitz determinants for a subclass of q-starlike functions associated with a general conic domain. Mathematics 2019, 7, 181. [Google Scholar] [CrossRef]
8. Srivastava, H.M.; Tahir, M.; Khan, B.; Ahmad, Q.Z.; Khan, N. Some general classes of q-starlike functions associated with the Janowski functions. Symmetry 2019, 11, 292. [Google Scholar] [CrossRef]
9. Mahmood, S.; Jabeen, M.; Malik, S.N.; Srivastava, H.M.; Manzoor, R.; Riaz, S.M.J. Some coefficient inequalities of q-starlike functions associated with conic domain defined by q-derivative. J. Funct. Spaces 2018, 2018, 8492072. [Google Scholar] [CrossRef]
10. Srivastava, H.M.; Bansal, D. Close-to-convexity of a certain family of q-Mittag-Leffler functions. J. Nonlinear Var. Anal. 2017, 1, 61–69. [Google Scholar]
11. Aldweby, H.; Darus, H. Some subordination results on q-analogue of Ruscheweyh differential operator. Abstr. Appl. Anal. 2014, 2014, 958563. [Google Scholar] [CrossRef]
12. Ezeafulukwe, U.A.; Darus, M. A note on q-calculus. Fasc. Math. 2015, 55, 53–63. [Google Scholar] [CrossRef]
13. Ezeafulukwe, U.A.; Darus, M. Certain properties of q-hypergeometric functions. Int. J. Math. Math. Sci. 2015, 2015, 489218. [Google Scholar] [CrossRef]
14. Sahoo, S.K.; Sharma, N.L. On a generalization of close-to-convex functions. Ann. Pol. Math. 2015, 113, 93–108. [Google Scholar] [CrossRef] [Green Version]
15. Uçar, H.E.Ö. Coefficient inequality for q-starlike functions. Appl. Math. Comput. 2016, 276, 122–126. [Google Scholar]
16. Kanas, S.; Răducanu, D. Some class of analytic functions related to conic domains. Math. Slovaca 2014, 64, 1183–1196. [Google Scholar] [CrossRef] [Green Version]
17. Ruscheweyh, S. New criteria for univalent functions. Proc. Am. Math. Soc. 1975, 49, 109–115. [Google Scholar] [CrossRef]
18. Janteng, J.; Abdulhalim, S.; Darus, M. Hankel determinant for starlike and convex functions. Int. J. Math. Anal. 2007, 1, 619–625. [Google Scholar]
19. Mishra, A.K.; Gochhayat, P. Second Hankel determinant for a class of analytic functions defined by fractional derivative. Int. J. Math. Math. Sci. 2008, 2008, 1–10. [Google Scholar] [CrossRef]
20. Singh, G.; Singh, G. On the second Hankel determinant for a new subclass of analytic functions. J. Math. Sci. Appl. 2014, 2, 1–3. [Google Scholar]
21. Srivastava, H.M.; Altinkaya, Ş.; Yalçın, S. Hankel determinant for a subclass of bi-univalent functions defined by using a symmetric q-derivative operator. Filomat 2018, 32, 503–516. [Google Scholar] [CrossRef]
22. Babalola, K.O. On $H 3 1$ Hankel determinant for some classes of univalent functions. Inequal. Theory Appl. 2007, 6, 1–7. [Google Scholar]
23. Ma, W.C.; Minda, D.A. unified treatment of some special classes of univalent functions. In Proceedings of the Conference on Complex Analysis (Tianjin, 1992); Li, Z., Ren, F., Yang, L., Zhang, S., Eds.; International Press: Cambridge, UK, 1994; pp. 157–169. [Google Scholar]
24. Libera, R.J.; Zlotkiewicz, E.J. Early coefficient of the inverse of a regular convex function. Proc. Am. Math. Soc. 1982, 85, 225–230. [Google Scholar] [CrossRef]
25. Libera, R.J.; Zlotkiewicz, E.J. Coefficient bounds for the inverse of a function with derivative in $𝒫$. Proc. Am. Math. Soc. 1983, 87, 251–257. [Google Scholar] [CrossRef]
26. Duren, P.L. Univalent Functions; Grundlehren der Mathematischen Wissenschaften, Band 259; Springer: New York, NY, USA; Berlin/Heidelberg, Germany; Tokyo, Japan, 1983. [Google Scholar]

## Share and Cite

MDPI and ACS Style

Mahmood, S.; Srivastava, H.M.; Khan, N.; Ahmad, Q.Z.; Khan, B.; Ali, I. Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions. Symmetry 2019, 11, 347. https://doi.org/10.3390/sym11030347

AMA Style

Mahmood S, Srivastava HM, Khan N, Ahmad QZ, Khan B, Ali I. Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions. Symmetry. 2019; 11(3):347. https://doi.org/10.3390/sym11030347

Chicago/Turabian Style

Mahmood, Shahid, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Bilal Khan, and Irfan Ali. 2019. "Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions" Symmetry 11, no. 3: 347. https://doi.org/10.3390/sym11030347

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