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

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Citation for this paper:

Mahmood, S., Srivastava, H.M., Khan, N., Ahmad, Q.Z., Khan, B. & Ali, I. (2019).

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

Functions. Symmetry, 11(3), 347.

https://doi.org/10.3390/sym11030347

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

Shahid Mahmood, Hari M. Srivastava, Nazar Khan, Qazi Zahoor Ahmad, Bilal Khan

and Irfan Ali

March 2019

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open

access article distributed under the terms and conditions of the Creative Commons

Attribution (CC BY) license (

http://creativecommons.org/licenses/by/4.0/

).

This article was originally published at:

https://doi.org/10.3390/sym11030347

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symmetry

S S

Article

Upper Bound of the Third Hankel Determinant for a

Subclass of q-Starlike Functions

Shahid Mahmood1, Hari M. Srivastava2,3 , Nazar Khan4, Qazi Zahoor Ahmad4,* , Bilal Khan4 _{and Irfan Ali}5

1 _{Department of Mechanical Engineering, Sarhad University of Science and Information Technology,} Ring Road, Peshawar 25000, Pakistan; shahidmahmood757@gmail.com

2 _{Department of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, Canada;} harimsri@math.uvic.ca

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;} nazarmaths@gmail.com (N.K.); bilalmaths789@gmail.com (B.K.)

5 _{Department of Mathematical Sciences, Balochistan University of Information Technology, Engineering and} Management Sciences, Quetta 87300, Pakistan; irfan.ali@buitms.edu.pk

* Correspondence: zahoorqazi5@gmail.com; Tel.: +92-334-96-60-162

Received: 1 January 2019; Accepted: 23 February 2019; Published: 7 March 2019

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.

Keywords:analytic functions; Hadamard product; starlike functions; q-derivative (or q-difference) operator; Hankel determinant; q-starlike functions

MSC:Primary 05A30, 30C45; Secondary 11B65, 47B38

1. Introduction

We denote byA (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 anzn (∀z∈ U) (1)

in the open unit diskU, centered at the origin and normalized by the conditions given by

f(0) =0 and f0(0) =1.

In addition, letS ⊂ Abe the class of functions which are univalent inU. The class of starlike

functions inUwill be denoted byS∗, which consists of normalized functions f ∈ Athat satisfy the

following inequality: < z f 0₍_z₎ f(z) >0, (∀z∈ U). (2)

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If two functions f and g are analytic inU, 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 inU, with

w(0) =0 and |w(z)| <1, such that

f(z) =g w(z).

In particular, if the function g is univalent inU, 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 by

f(z) =z+ ∞

∑

n=2 anzn (∀z∈ U) and g(z) =z+ ∞

∑

n=2 bnzn (∀z∈ U),

the convolution (or the Hadamard product) of f and g is defined as follows: f(z) ∗g(z) =z+

∞

∑

n=2

anbnzn.

We next denote byPthe class of analytic functions p which are normalized by p(z) =1+ ∞

∑

n=1 pnzn, (3) such that < 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,· · · } = N0\ {0} (N0={0, 1, 2, 3,· · · }).

Definition 1. Let q∈ (0, 1)and define the q-number[λ]_qby

[λ]_q=            1−qλ 1−q (λ∈ C) n−1 ∑ k=0 qk ₌₁₊_q₊_q2_{+ · · · +}_qn−1 ₍_λ₌_n_{∈ N)}_.

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Definition 2. Let q∈ (0, 1)and define the q-factorial[n]_q! by [n]_q!=        1 (n=0) n−1 ∏ k=1 [k]_q (n∈ N).

Definition 3. Let q∈ (0, 1)and define the generalized q-Pochhammer symbol[λ]_q,nby

[λ]_q,n=        1 (n=0) n ∏ k=0 [λ+k]_q (n∈ N).

Definition 4. For ω>0, let the q-gamma functionΓq(ω)be defined by Γq(ω+1) = [ω]_qΓq(ω) and Γq(1):=1.

Definition 5. (see [3,4]) The q-derivative (or the q-difference) operator Dqof a function f in a given subset of

Cis defined by Dqf(z) =        f(z) − f(qz) (1−q)z (z6=0) f0(0) (z=0), (4)

provided that f0(0)exists.

We note from Definition5that lim q→1− Dqf (z) = lim q→1− f(qz) − f(z) (1−q)z = f 0 (z),

for a differentiable function f in a given subset ofC. It is readily deduced from(1)and(4)that

Dqf(z) =1+ ∞

∑

n=2

[n]_qanzn−1. (5)

The operator Dqplays a vital role in the investigation and study of numerous subclasses of the

class of analytic functions of the form given in Definition5. A q-extension of the class of starlike functions was first introduced in [5] by using the q-derivative operator (see Definition6 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 Dqin analytic function theory can be found in [7–13] 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) = f0(0) −1=0 (6) and z f(z) Dqf (z) − 1 1−q 5 1 1−q (∀z∈ U). (7) The notationS∗

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It is readily observed that, as q→1−, the closed disk given w− 1 1−q 5 1 1−q

becomes the right-half plane and the classS_q∗reduces toS∗. 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) Dqf (z) ≺p_b b p= 1+z 1−qz .

Definition 7. (see [16]) For a function f ∈ A (U), the Ruscheweyh-type q-derivative operator is defined as follows: Rδ qf(z) =φ(q, δ+1; z) ∗f(z) =z+ ∞

∑

n=2 ψn−1anzn (z∈ U; δ> −1), (8) where φ(q, δ+1; z) =z+ ∞

∑

n=2 ψn−1zn (9) and ψn−1= Γq(δ+n) [n−1]_q!Γq(δ+1) = [n+1]_n−1,q [n−1]_q! . (10) From (8) it can be seen that

R0 qf(z) = f(z) and Rq1f(z) =zDqf(z), Rm_q f(z) = zD m q f(z) zm−1f(z) [m]_q! (m∈ N), lim q→1−φ(q, δ+1; z) = z (1−z)δ+1 and lim q→1−R δ qf(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 Dδ_f₍_z₎ _{(see [}₁₇_{]). From (}₈_{) the following identity can} easily be derived: zDqRδqf(z) = 1+ [δ]_q qδ ! Rδ+1 q f(z) − [δ]_q qδ R δ qf(z). (11) If q→1−, then zRδ_f₍_z₎0 _{= (}₁₊_δ_{) R}δ+1_f₍_z_{) −}_δ_Rδ_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 classRS∗_q(δ)if the following inequality holds true: zDqRδqf(z) f(z) − 1 1−q 5 1 1−q (z∈ U; δ> −1)

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or, equivalently, we have (see [15])

zDqRδqf(z)

f(z) ≺

1+z

1−qz (12)

by using the principle of subordination.

Let n=0 and j=1. The jth Hankel determinant is defined as follows:

Hj(n) = an an+1 . . . an+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 onH2(2)were obtained by several authors (see, for example, [18–21]) for various classes of

normalized analytic functions. It is well-known that the Fekete-Szegö functional a3−a22

= H₂(1). This functional is further generalized asa₃−µa2₂

for some real or complex µ. In fact, Fekete and Szegö gave sharp estimates ofa₃−µa2₂

for real µ and f ∈ S, the class of normalized univalent functions inU. It is also known that the functionala2a4−a23

is equivalent toH2(2). Babalola [22] studied the

Hankel determinantH3(1)for some subclasses of analytic functions. In the present investigation, our

focus is on the Hankel determinantH3(1)for the above-defined function classRS∗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+c1z+c2z2+ · · ·

be in the classPof functions with positive real part inU. Then, for any complex number υ,

c2−υc 2 1 5              −4υ+2 (υ50) 2 (05υ51) 4υ−2 (υ=1). (13)

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−z2

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 (05ρ51)

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.

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Lemma 2. (see [24,25]) Let

p(z) =1+p1z+p2z2+ · · ·

be in the classPof functions with positive real part inU. Then

2p2=p21+x

4−p2₁ for some x,|x| 51 and

4p3= p31+2 4−p2₁p1x− 4−p2₁p1x2+2 4−p2₁ 1− |x|2z for some z (|z| 51).

Lemma 3. (see [26]) Let

p(z) =1+p1z+p2z2+ · · ·

be in the classPof functions positive real part inU. Then

|pk| 52 (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 ∈ RS_q∗(δ)be of the form(1). Then

a3−µa 2 2 5                              1+q+q2 ψ₁2−µ(1+q)2ψ2 q2_ψ 2ψ2₁ µ< q 2₊₁ ψ2₁ (1+q)2ψ2 ! 1 qψ2 q2+1 ψ2₁ (1+q)2ψ2 5µ5 ψ 2 1 ψ2 ! µ(1+q)2ψ2− 1+q+q2ψ₁2 q2_ψ 2ψ2₁ µ > ψ 2 1 ψ2 ! , where ψn−1is given by(10).

It is also asserted that, for

q2₊₁ ψ₁2 (1+q)2ψ2 5µ5 1 +q+q2 ψ₁2 (1+q)2ψ2 , |a3−µa2₂| + µ− q 2₊₁ ψ2₁ (1+q)2ψ2 ! |a2|2≤ 1 qψ2

and that , for

1+q+q2 ψ2₁ (1+q)2ψ2 5µ5 ψ 2 1 ψ2 ,

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|a3−µa2₂| + ψ 2 1−µψ2 ψ2 ! |a2|25 1 qψ2 .

Proof. If f ∈ RS∗_q(δ), then it follows from(12)that

zDqRδqf(z) f(z) ≺φ(z), (14) where φ(z) = 1+z 1−qz. We define a function p(z)by p(z) = 1+w(z) 1−w(z) =1+p1z+p2z 2₊_p 3z3+ · · ·.

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

zDqRδqf(z) f(z) =φ w(z), together with φ(w(z)) = 2p(z) (1−q)p(z) +1+q. Now 2p(z) (1−q)p(z) +1+q =1+1 2(1+q)p1z+ 1 2(q+1)p2− 1 4(1−q 2₎_p2 1 z2 + 1 2(1+q)p3− 1 2(1−q 2₎_p 1p2+1 8(1+q)(1−q) 2_p3 1 z3 + ( 1 2(1+q)p4= 1 4 1−q2p2₂−1 2 1−q2p1p3 +3 8(1+q)(q−1) 2_p2 1p2+ 1 16(1+q)(1−q) 3_p4 1 ) z4+ · · ·. Similarly, we get zDqRδqf(z) Rδ qf(z) = 1+qa2ψ1z+ q+q2 ψ2a3−qψ21a22 z2+ ( q+q2₊_q3 ψ3a4 − 2q+q2 ψ1ψ2a2a3+qψ31a32 ) z3+ ( q+q2+q3+q4 ψ5a5 − 2q+q2+q3 ψ2ψ3a2a4− q+q2ψ2₂a2₃ + 3q+q2 ψ2₁ψ2a22a3−qψ₁4a42 ) z4+ · · ·,

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Therefore, we have a2= (1+q) 2qψ1 p1, (15) a3= 1 2qψ2p2 + q 2₊₁ 4q2_ψ 2 p2₁ (16) and a4 = (1+q) 2q(1+q+q2₎_ψ 3 p3− (1+q) (q−2) (2q+1) 4q2₍₁₊_q₊_q2₎_ψ 3 p1p2 +(1+q) q 2₊₁ q2₋_q₊₁ 8q3₍₁₊_q₊_q2₎_ψ 3 p3 1. (17) We thus obtain a3−µa 2 2 = 1 2qψ2 p2− µ(1+q)2ψ2− 1+q2ψ2₁ 2qψ2₁ ! p2₁ . (18)

Finally, by applying Lemma1and Equation (13) in conjunction with (18), we obtain the result asserted by Theorem1.

We now state and prove Theorem2below.

Theorem 2. Let f ∈ RS_q∗(δ)be of the form(1). Then

a2a4−a 2 3 5 1 q2_ψ2 2 .

Proof. From (15)–(17), we obtain

a2a4−a23= (1+q)2 4q2₍₁₊_q₊_q2₎_ψ 1ψ3 ! p1p3− (1+q)2(q−2) (2q+1) 8q3₍₁₊_q₊_q2₎_ψ 1ψ3 + q 2₊₁ 4q3_ψ2 2 ! p2₁p − 1 4q2_ψ2 2 ! p2 2+ − q2₊₁2 16q4_ψ2 2 +(1+q) 2 _q2₊₁ q2₋_q₊₁ 16q3₍₁₊_q₊_q2₎_ψ 1ψ3 ! p4 1.

By using Lemma2, we have

a2a4−a23= (1+q)2 q2+1 q2−q+1 16q3₍₁₊_q₊_q2₎_ψ 1ψ3 − q 2₊₁2 16q4_ψ2 2 ! p41 + (1+q) 2 16q2₍₁₊_q₊_q2₎_ψ 1ψ3 ! p1 n p3₁+2p1 4−p2₁x −p1 4−p2₁x2+24−p2₁ 1− |x|2zo+ q 2₊₁ 8q3_ψ2 2 ·(1+q) 2 (q−2) (2q+1) 16q3₍₁₊_q₊_q2₎_ψ 1ψ3 ! p2₁np2₁+4−p2₁xo − 1 16q2_ψ2 2 ! p4₁+4−p2₁2x2+2p2₁4−p2₁x .

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Now, taking the moduli and replacing|x|by ρ and p1by p, we have a2a4−a 2 3 5 1 Λ(q) h ω(q)p4+2q(1+q)2ψ₂2p 4−p2 +Ω(q)4−p2p2ρ+ q(q+1)2ψ₂2p2+q 4−p2 ·1+q+q2ψ1ψ3−2q(1+q)2ψ₂2p 4−p2ρ2 i =F(p, ρ), (19) where Λ(q) =16q31+q+q2ψ1ψ3ψ2₂, ω(q) = 3+3q−q3+q4(1+q)2ψ₂2− 1+3q+2q2+2q3+q4 ·1+q+q2ψ1ψ3 and Ω(q) = (1+q) 2 2q2−5q−2ψ2₂+2q q2+2 1+q+q2ψ1ψ3 .

Upon differentiating both sides(19)with respect to ρ, we have ∂F(p, ρ) ∂ρ = 1 Λ(q) hΩ(q)4−p2p2+2q(q+1)2ψ₂2p2+q 4−p2 ·1+q+q2ψ1ψ3−2q(1+q)2ψ2₂p 4−p2ρ i . 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) _h ω(q) −Ω(q) −q(q+1)2ψ2₂+ q+q2+q3ψ1ψ3 p4 +4Ω(q) +4q(q+1)2ψ2₂−8 q+q2+q3ψ1ψ3 p2 +16q+q2+q3ψ1ψ3 =G(p). (20)

By differentiating both sides of(20)with respect to p, we have G0(p) = 1 Λ(q) h 4ω(q) −Ω(q) −q(q+1)2ψ2₂+ q+q2+q3ψ1ψ3 p3 +24Ω(q) +4q(q+1)2ψ₂2−8 q+q2+q3ψ1ψ3 pi.

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Differentiating the above equation once again with respect to p, we get G00(p) = 1 Λ(q) h 12ω(q) −Ω(q) −q(q+1)2ψ₂2+ q+q2+q3ψ1ψ3 p2 +24Ω(q) +4q(q+1)2ψ2₂−8 q+q2+q3ψ1ψ3 i . For p=0, this shows that the maximum value of(G(p))occurs at p=0. Hence, we obtain

a2a4−a 2 3 5 1 q2_ψ2 2 . The proof of Theorem2is thus completed.

If, in Theorem2, 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 a2a4−a 2 3 51, and the inequality is sharp.

Theorem 3. Let f ∈ RS∗_q(δ). Then

|a2a3−a4| 5 (1+q)κ(q) ψ1ψ2ψ3(q2+q3+q4) , where κ(q) = 1+q+q22ψ3− q4−3q+6q2+q+1ψ1ψ2 . (21)

Proof. Using the values given in(15)and(16)we have

a2a3−a4 = (1+q) q2₊₁ 8q3_ψ 1ψ2 −(1+q) q 2₊₁ q2₋_q₊₁ 8ψ3(q2+q3+q4) ! p3₁ + (1+q) 4q2_ψ 1ψ2 −(q−2) (2q+1) (1+q) 4ψ3(q2+q3+q4) p1p2 − (1+q) 2(q+q2₊_q3₎_ψ 3 p3. (22)

We now use Lemma2and assume that p1 5 2. In addition, by Lemma3, we let p1 = 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 obtain

|a2a3−a4| 5 (1+q) 8(q3₊_q4₊_q5₎_ψ 1ψ2ψ3 _h κ(q)p3+η(q)p(4−p2)ρ +2q2ψ1ψ2(4−p2) +q2ψ1ψ2(p−2) (4−p2)ρ2 i =: F(ρ), where η(q) = q+q2+q3ψ3+ 2q3−q2−2qψ1ψ2

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and κ(q)is given by(21). Differentiating F(ρ)with respect to ρ, we have F0(ρ) = (1+q) 8(q3₊_q4₊_q5₎_ψ 1ψ2ψ3 h η(q)p(4−p2) +2q2ψ1ψ2(p−2) (4−p2)ρ i >0.

This implies that F(ρ)is an increasing function of ρ on the closed interval[0, 1]. Hence, we have F(ρ) 5F(1) (∀ρ∈ [0, 1]), that is, F(ρ) 5 (1+q) 8(q3₊_q4₊_q5₎_ψ 1ψ2ψ3 h κ(q) −η(q) −q2ψ1ψ2 p3 +4η(q) +4q2ψ1ψ2 pi =: G(p).

Since p∈ [0, 2], p=2 is a point of maximum. We thus obtain G(p)5 (1+q)κ(q)

(q3₊_q4₊_q5₎_ψ 1ψ2ψ3

, 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 Theorem3.

Corollary 2. (see [22]) Let f ∈S∗. Then

|a2a3−a4| 52.

Finally, we prove Theorem4below.

Theorem 4. Let f ∈ RS∗_q(δ). Then

H3(1) 5 " 1+q+q2 q4_ψ3 2 + κ (q)κ(q) q5₍₁₊_q₊_q2₎2_ψ 1ψ2ψ₃2 + τ(q) q5₍₁₊_q₊_q2₊_q3_{) (}₁₊_q₊_q2₎_ψ 2ψ4 # , where κ (q) = (1+q)2 q4−3q3+6q2+q+1, (23) τ(q) = (1+q) 4q7+2q6+6q5+7q4+13q3−q−1 (24) and κ(q)is given by(21). Proof. Since H₃(1) 5 |a3| a2a4−a 2 3 +|a4| |a2a3−a4| + |a5| a3−a 2 2 ,

by using Lemma3, we have

|a4| 5

(1+q) 1+q+6q2−3q3+q4 q3₍₁₊_q₊_q2₎_ψ

(13)

and |a5| 5 τ (q) q4₍₁₊_q₊_q2₊_q3_{) (}₁₊_q₊_q2₎_ψ 4 ,

where τ(q)is given by(24). Now, by applying Theorems1–3, we have the required result asserted by Theorem4.

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 diskU. 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 Theorems1–4. 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

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