# Hankel Determinant forp-Valent Alpha-Convex Functions

Authors: Gagandeep Singh; B. S. Mehrok
Journal: Geometry
 (2013)

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## Abstract

The objective of the present paper is to obtain the sharp upper bound of|ap+1ap+3−ap+22| for p-valent α-convex functions of the form f(z)=zp+∑k=p+1∞akzk in the unit disc E={z:|z|<1}.


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## Body

## 1. Introduction

LetAp be the class of  analytic  functions of the form
(1)f(z)=zp+∑k=p+1∞akzk
in the unit disc E={z:|z|<1} with p∈N={1,2,3,…}. Let S be the subclass of A1=A, consisting of  univalent functions.S
p
* is the class consisting of functions of the form (1) and satisfying the condition
(2)Re{zf′(z)pf(z)}>0,z∈E.The functions of the classSp* are called p-valent starlike functions. In particular, S1*≡S*, the class of starlike functions.K
p is the class of functions of the form (1), satisfying the condition
(3)Re{(zf′(z))′pf′(z)}>0,z∈E.The functions of the classKp are known as p-valent convex functions. Particularly, K1≡K, the class of convex functions.Obviouslyf(z)∈Kp if and only if zf′(z)/p∈Sp*.LetMp(α)(α≥0) be the class of functions of the form (1), satisfying the condition
(4)Re{(1-α)zf′(z)pf(z)+α(zf′(z))′pf′(z)}>0,z∈E.Functions in the classMp(α) are known as p-valent alpha-convex functions. For p=1, the class Mp(α) reduces to the class M(α) of alpha-convex functions introduced by Mocanu [1]. Also Mp(0)≡Sp* and Mp(1)≡Kp.In 1976, Noonan and Thomas [2] stated the qth  Hankel determinant for q≥1 and n≥1 as
(5)Hq(n)=|anan+1⋯an+q-1an+1⋯⋯⋯⋯⋯⋯⋯an+q-1⋯⋯an+2q-2|.This determinant   has   also   been    considered   by   several   authors.  For   example, Noor [3] determined the rate of growth of Hq(n) as n→∞ for functions given by (1) with bounded boundary. Ehrenborg [4] studied  the  Hankel determinant of exponential polynomials. Also Hankel determinant was  studied by various authors including Hayman [5] and Pommerenke [6]. In [7], Janteng et al.  studied the Hankel determinant for the classes of starlike and convex functions. Again  Janteng et al. discussed the Hankel determinant problem for the classes of starlike functions with respect to symmetric points and convex functions with respect to symmetric points in [8] and for the functions whose derivative has a positive real part in [9]. Also Hankel determinant for various subclasses of  p-valent functions was investigated by various authors including Krishna and Ramreddy [10] and Hayami and Owa [11].Easily, one can observe that the Fekete and Szegö functional  isH2(1). Fekete and Szegö [12] then further generalised the estimate |a3-μa22|, where μ is real and f∈S. For our discussion in this paper, we consider the Hankel determinant in the case of q=2 and n=2:
(6)|a2a3a3a4|.In this paper, we seek sharp upper bound of the functional|ap+1ap+3-ap+22| for functions belonging to the class Mp(α). The results due to Janteng et al. [7] follow as special cases.


## 2. Preliminary Results

LetQ be the family of all functions q analytic in E  for which Re(q(z))>0 and
(7)q(z)=1+q1z+q2z2+⋯
for z∈E.Lemma 1 (see [6]).
Ifq∈Q, then |qk|≤2(k=1,2,3,…).Lemma 2 (see [13, 14]).
Ifq∈Q, then
(8)2q2=q12+(4-q12)x,4q3=q13+2q1(4-q12)x-q1(4-q12)x2+2(4-q12)(1-|x|2)z,
for some x and z satisfying |x|≤1,|z|≤1 and  q1∈[0,2].


## 3. Main Result

Theorem 3.
Iff∈Mp(α), then
(9)|ap+1ap+3-ap+22|≤p4(p+2α)2[1-12α2(p+α)4(p+3α)A(α)],
where
(10)A(α)=-24p(p+2α)[α(p+1)(p+2)-(α-1)p2]×[(p+α)2+p(p2+α+2αp)]+12p(p+3α)(p2+α+2αp)×[p(p2+α+2αp)+2(p+α)2]+4(p+α)3[3(p+α)(p+3α)-4(p+2α)2]-16p2(p+2α)2[(α-1)p3-α(p+1)3].Proof .
Sincef∈Mp(α), so from (4)
(11)(1-α)zf′(z)pf(z)+α(zf′(z))′pf′(z)=q(z).
On expanding and equating the coefficients of z,z2, and z3 in  (11), we get
(12)ap+1=p2q1p+α,ap+2=p2q22(p+2α)+p3(p2+α+2αp)q122(p+2α)(p+α)2,ap+3=p2q33(p+3α)+[3α(p+1)(p+2)-3(α-1)p2]p3q1q26(p+α)(p+2α)(p+3α)+p4[(p2+α+2αp)×(3α(p+1)(p+2)-3(α-1)p2)+2(p+2α)((α-1)p3-α(p+1)3)]q13×(6(p+2α)(p+3α)(p+α)3)-1.
Equation (12) yields:
(13)ap+1ap+3-ap+22=p4C(α)×{q224(p+2α)2(p+α)3q1q3+[2p(p+2α)(p+α)2×(3α(p+1)(p+2)-3(α-1)p2)-6p(p+3α)(p+α)2×(p2+α+2αp)]q12q2+[2p2(p+2α)×((p2+α+2αp)×(3α(p+1)(p+2)-3(α-1)p2)+2(p+2α)((α-1)p3-α(p+1)3))-3p2(p+3α)(p2+α+2αp)2]q14-3(p+3α)(p+α)4q22},where  C(α)=12(p+3α)(p+2α)2(p+α)4.
Using Lemmas1 and 2 in (13), we obtain
(14)|ap+1ap+3-ap+22|=p44C(α)×|-[-4(p+2α)2(p+α)3+3(p+α)4×(p+3α)-4p(p+2α)(p+α)2×(3α(p+1)(p+2)-3(α-1)p2)+12p(p+3α)(p+α)2(p2+α+2αp)+12p2(p+3α)(p2+α+2αp)2-8p2(p+2α)(p2+α+2αp)×[3α(p+1)(p+2)-3(α-1)p2]-16p2(p+2α)2[(α-1)p3-α(p+1)3]]q14+(p+α)2[8(p+2α)2(p+α)+4p(p+2α)(p+α)2×(3α(p+1)(p+2)-3(α-1)p2)(p+α)2-6(p+α)2(p+3α)-12p(p+3α)(p+α)2×(p2+α+2αp)]q12(4-q12)x-(p+α)3[4(p+2α)2q12+3(p+α)(p+3α)×(4-q12)](4-q12)x2+8(p+α)3(p+2α)2q1(4-q12)(1-|x|2)z|.
Assume that q1=q and q∈[0,2]; using triangular inequality and |z|≤1, we have
(15)|ap+1ap+3-ap+22|≤p44C(α)×{[-4(p+2α)2(p+α)3+3(p+α)4×(p+3α)-4p(p+2α)(p+α)2×(3α(p+1)(p+2)-3(α-1)p2)+12p(p+3α)(p+α)2(p2+α+2αp)+12p2(p+3α)(p2+α+2αp)2-8p2(p+2α)(p2+α+2αp)×[3α(p+1)(p+2)-3(α-1)p2]-16p2(p+2α)2[(α-1)p3-α(p+1)3]]q4+(p+α)2[8(p+2α)2(p+α)+4p(p+2α)×(3α(p+1)(p+2)-3(α-1)p2)-6(p+α)2(p+3α)-12p(p+3α)×(p2+α+2αp)]q2(4-q2)δ+8(p+α)3(p+2α)2q(4-q2)+(p+α)3[(4(p+2α)2-3(p+α)(p+3α))q2-8(p+2α)2q+12(p+α)×(p+3α)(p+2α)2](4-q2)δ2}=p44C(α)F(δ),whereδ=|x|≤1.
It is easy to verify that  F(δ) is an increasing function and so  Max.F(δ)=F(1).
Consequently(16)|ap+1ap+3-ap+22|≤p44C(α)G(q),
where
(17)G(q)=F(1).
So
(18)G(q)=A(α)q4+48α(p+α)4q2+48(p+3α)(p+α)4,
where  A(α)  is defined in (10).
Now(19)G′(q)=4A(α)q3+96α(p+α)4q,G′′(q)=12A(α)q2+96α(p+α)4.G′(q)=0  gives
(20)4q[A(α)q2+24α(p+α)4]=0.G′′(q)  is negative at   q=-24α(p+α)4/A(α)=q′.
So(21)Max.G(q)=G(q′).
Hence from (15), we obtain (9).The result is sharp forq1=q′, q2=q12-2,  and q3=q1(q12-3).Forα=0, Theorem 3 gives the following result.Corollary 4.
Iff(z)∈Sp*, then
(22)|ap+1ap+3-ap+22|≤p2.Forα=1, Theorem 3 yields.Corollary 5.
Iff(z)∈Kp, then
(23)|ap+1ap+3-ap+22|≤p4(p+2)2[1+3(p+3)(-p3-3p2+3p+7)].Puttingα=0 and p=1  in Theorem 3, we obtain the following result due to Janteng et al. [7].Corollary 6.
Iff(z)∈S*, then
(24)|a2a4-a32|≤1.Puttingα=1 and p=1  in Theorem 3, we obtain the following result due to Janteng et al. [7].Corollary 7.
Iff(z)∈K, then
(25)|a2a4-a32|≤18.



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*Source: 348251-2013-10-08.xml*