GEOMETRY Geometry 2314-4238 2314-422X Hindawi Publishing Corporation 348251 10.1155/2013/348251 348251 Research Article Hankel Determinant for p-Valent Alpha-Convex Functions Singh Gagandeep Mehrok B. S. Saadati Reza Department of Mathematics M.S.K. Girls College Bharowal (Tarn-Taran), Punjab 143401 India 2013 8 10 2013 2013 14 06 2013 29 08 2013 2013 Copyright © 2013 Gagandeep Singh and B. S. Mehrok. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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

1. Introduction

Let Ap be the class of analytic functions of the form (1)f(z)=zp+k=p+1akzk in the unit disc E={z:|z|<1} with pN={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,zE.

The functions of the class Sp* 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,zE.

The functions of the class Kp are known as p-valent convex functions. Particularly, K1K, the class of convex functions.

Obviously f(z)Kp if and only if zf(z)/pSp*.

Let Mp(α)(α0) 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,zE.

Functions in the class Mp(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 . Also Mp(0)Sp* and Mp(1)Kp.

In 1976, Noonan and Thomas  stated the qth Hankel determinant for q1 and n1 as (5)Hq(n)=|anan+1an+q-1an+1an+q-1an+2q-2|.

This determinant has also been considered by several authors. For example, Noor  determined the rate of growth of Hq(n) as n for functions given by (1) with bounded boundary. Ehrenborg  studied the Hankel determinant of exponential polynomials. Also Hankel determinant was studied by various authors including Hayman  and Pommerenke . In , 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  and for the functions whose derivative has a positive real part in . Also Hankel determinant for various subclasses of p-valent functions was investigated by various authors including Krishna and Ramreddy  and Hayami and Owa .

Easily, one can observe that the Fekete and Szegö functional is H2(1). Fekete and Szegö  then further generalised the estimate |a3-μa22|, where μ is real and fS. 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.  follow as special cases.

2. Preliminary Results

Let Q be the family of all functions q analytic in E for which Re(q(z))>0 and (7)q(z)=1+q1z+q2z2+ for zE.

Lemma 1 (see [<xref ref-type="bibr" rid="B14">6</xref>]).

If qQ, then |qk|2  (k=1,2,3,).

Lemma 2 (see [<xref ref-type="bibr" rid="B9">13</xref>, <xref ref-type="bibr" rid="B10">14</xref>]).

If qQ, 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.

If fMp(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 .

Since fMp(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 Lemmas 1 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),where  δ=|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 for q1=q, q2=q12-2, and q3=q1(q12-3).

For α=0, Theorem 3 gives the following result.

Corollary 4.

If f(z)Sp*, then (22)|ap+1ap+3-ap+22|p2.

For α=1, Theorem 3 yields.

Corollary 5.

If f(z)Kp, then (23)|ap+1ap+3-ap+22|p4(p+2)2[1+3(p+3)(-p3-3p2+3p+7)].

Putting α=0 and p=1 in Theorem 3, we obtain the following result due to Janteng et al. .

Corollary 6.

If f(z)S*, then (24)|a2a4-a32|1.

Putting α=1 and p=1 in Theorem 3, we obtain the following result due to Janteng et al. .

Corollary 7.

If f(z)K, then (25)|a2a4-a32|18.

Mocanu P. T. Une propriété de convexité généralisée dans la théorie de la représentation conforme Mathematica 1969 11 34 127 133 ZBL0195.36401 MR0273000 Noonan J. W. Thomas D. K. On the second Hankel determinant of areally mean p-valent functions Transactions of the American Mathematical Society 1976 223 337 346 ZBL0346.30012 MR0422607 Noor K. I. Hankel determinant problem for the class of functions with bounded boundary rotation Académie de la République Populaire Roumaine. Revue Roumaine de Mathématiques Pures et Appliquées 1983 28 8 731 739 ZBL0524.30008 MR725316 Ehrenborg R. The Hankel determinant of exponential polynomials The American Mathematical Monthly 2000 107 6 557 560 10.2307/2589352 MR1767065 ZBL0985.15006 Hayman W. K. Multivalent Functions 1958 48 Cambridge, UK Cambridge University Press Cambridge Tracts in Mathematics and Mathematical Physics MR0108586 Pommerenke Ch. Univalent Functions 1975 Göttingen Vandenhoeck & Ruprecht MR0507768 Janteng A. Halim S. A. Darus M. Hankel determinant for starlike and convex functions International Journal of Mathematical Analysis 2007 1 13-16 619 625 MR2370200 Janteng A. Halim S. A. Darus M. Hankel determinant for functions starlike and convex with respect to symmetric points Journal of Quality Measurement and Analysis 2006 2 1 37 43 Janteng A. Halim S. A. Darus M. Coefficient inequality for a function whose derivative has a positive real part Journal of Inequalities in Pure and Applied Mathematics 2006 7 2, article 50 MR2221331 Krishna D. V. Ramreddy T. Hankel determinant for p-valent starlike and convex functions of order α Novi Sad Journal of Mathematics 2012 42 2 89 96 Hayami T. Owa S. Hankel determinant for p-valently starlike and convex functions of order α General Mathematics 2009 17 4 29 44 MR2733349 Fekete M. Szegö G. Eine Bemerkung über ungerade schlichte Funktionen Journal of the London Mathematical Society 1933 8 85 89 ZBL0006.35302 Libera R. J. Złotkiewicz E. J. Early coefficients of the inverse of a regular convex function Proceedings of the American Mathematical Society 1982 85 2 225 230 ZBL0464.30019 10.2307/2044286 MR652447 Libera R. J. Złotkiewicz E. J. Coefficient bounds for the inverse of a function with derivative in P Proceedings of the American Mathematical Society 1983 87 2 251 257 ZBL0488.30010 10.2307/2043698 MR681830