MSC 2010: 30C45, 30C50The object of this paper is to obtain sharp results involving coefficient bounds, growth and distortion properties for some classes of analytic and univalent functions with negative coefficients

Pashkouleva, Donka

Fractional Calculus and Applied Analysis

Bulgarian Digital Mathematics Library at IMI-BAS

SOME NOTES ABOUT A CLASS OF UNIVALENTFUNCTIONS WITH NEGATIVE COEFFICIENTSDonka PashkoulevaThis paper is dedicated to the 70th anniversary of Professor SrivastavaAbstractThe object of this paper is to obtain sharp results involving coefficientbounds, growth and distortion properties for some classes of analytic andunivalent functions with negative coefficients.MSC 2010: 30C45, 30C50Key Words and Phrases: univalent functions with negative coefficients,radius of convexity1. Introduction and definitionsLet S denote the class of functions of the form:f(z) = z +∞∑n=2anznthat are analytic and univalent in the unit disk E. We denote by C and S∗the classes of convex and starlike functions, respectively.A function f(z) analytic in E, is said to be starlike of order β (0 ≤ β < 1)in E if f(0) = f ′(0)− 1 = 0 and for z ∈ E<zf′(z)f(z)> βThe class of such functions will be denoted by S∗β. Clearly, S∗0 = S∗.c© 2010, FCAA – Diogenes Co. (Bulgaria). All rights reserved.416 D. PashkoulevaA function f(z) analytic in E is said to be close-to-convex of order β(0 ≤ β < 1) in E if there exists a function g(z) ∈ S∗ and a real number γsuch that, for z ∈ E and γ ∈(−pi2,pi2)<eiγ zf′(z)g(z)> β.The class of such functions is denoted by Kβ.A function f(z) is said to be close-to-star of order β (0 ≤ β < 1) if thereexists a function g(z) ∈ S∗ such that, for z ∈ E<f(z)g(z)> β.The class of such functions will be denoted by Rβ.A function f(z), analytic in E with f(0) = f ′(0) − 1 = 0 is said to bequasi-convex if and only if there exists a function g(z) ∈ C such that forz ∈ E<(zf′(z))′g′(z)> β.The class of such functions will be denoted by C∗β.Let T denotes the subclass of S, consisting of functions f(z) of the formf(z) = z −∞∑n=2|an|zn.We denote T ∗β = S∗β ∩ T ; K∗β = Kβ ∩ T ; R∗β = Rβ ∩ T ; L∗β = C∗β ∩ T .It is known that T = T ∗0 = T ∗ and f ∈ T ∗β if, and only if, for 0 ≤ β < 1∞∑n=2n− β1− β |an| ≤ 1 [1].In [2] Schild considered a subclass of T consisting of polynomials having|z| = 1 as a radius of univalence. Schild showed ([2]) that a necessary andsufficient condition for f ∈ T is1−∞∑n=2n|an| = 0.With the aid of this result he derived better results for certain quantitiesconnected with conformal mapping of univalent functions. Other subclassesof T have been studied by Gupta and Jain [1], [2] and Silverman [5], [6].In this paper we consider the following subclass Ht,α(β) of T :SOME NOTES ABOUT A CLASS OF UNIVALENT . . . 417Definition . A function f(z) = z−∞∑n=2|an|zn is said to be in Ht,α(β)(0 ≤ α < 1, 0 ≤ β < 1, 0 < t ≤ 1), if there exists a function g ∈ T ∗, withg(z) = z −∞∑n=2|bn|znsuch that for z ∈ E<{tzf ′(z) + (1− t)z (zf ′(z))′αg(z) + (1− α)zg′(z)}> β. (1)Evidently, H1,1(β) = K∗β, the class of close-to-convex functions of orderβ introduced by [7]. Note also that H1,0(β) = R∗β and H0,1(β) = L∗β.In the sequel we writeJt,α(f, g, z0) =11− β{tz0f′(z0) + (1− t)zo(zof ′(z0))′αg(z0) + (1− α)z0g′(z0) − β}. (2)2. Some results about the class Ht,α(β)Lemma . Let f ∈ Ht,α(β) be given by (1.1). Thenmin|z|≤r<1<Jt,α(f, g, z) = Jt,α(f, g, r).The proof of this lemma is standard.Theorem 1. Let f(z) ∈ Ht,α(β) be given by (1.1). Then, for 0 < r < 1,t∞∑n=2[n|an| − (α+ n(1− α))|bn|] rn−11−∞∑n=2(α+ n(1− α))|bn|rn−1++(1− t)∞∑n=2[n2|an| − (α+ n(1− α))|bn|]rn−11−∞∑n=2(α+ n(1− α))|bn|rn−1< 1− β(3)when 0 ≤ α ≤ 1, 0 ≤ t ≤ 1. The estimate (3) is also sufficient for f to be inHt,α(β).418 D. PashkoulevaRemark A. If∞∑n=2n|an| < 1,∞∑n=2n|bn| < 1 and∞∑n=2n2|an| < ∞, thenJt,α(f, g, r) is continuous at r = 1 and (2.1) may be replaced by∞∑n=2(tn+ (1− t)n2)|an| − β∞∑n=2(α+ (1− α)n) |bn| ≤ 1− β. (4)Remark B. In [5] it was shown that∞∑n=2|bn| ≤ 12 for g ∈ T∗, so that∞∑n=2(tn+ (1− t)n2)|an| ≤ 1− β + β(2− α)2 = 1−αβ2. (5)In fact, (5) is a necessary condition for f to be in Ht,α(β) and we couldalways take g(z) = z − 12z2 and (5) would also be sufficient.Theorem 2. Let f ∈ Ht,α(β) be given by (1.1). Thenan ≤ An = αβ + (1− αβ)nn2 [t+ (1− t)n] .The result is sharp for every n, with equality forf(z) = z −Anznand g ∈ T , with g(z) = z − 1nzn.Theorem 3. If f ∈ Ht,α(β)r − (2− αβ)4(2− t) r2 ≤ |f(z)| ≤ r + (2− αβ)4(2− t) r2, |z| ≤ r1− (2− αβ)2(2− t) r ≤ |f′(z)| ≤ 1 + (2− αβ)2(2− t) r, |z| ≤ r.Equality holds in all cases forf(z) = z − 2− αβ4(2− t)z2.3. Additional results for the class Ht,α(β)Theorem 4. The family Ht,α(β) is convex.SOME NOTES ABOUT A CLASS OF UNIVALENT . . . 419P r o o f. We modify the method of Silvia and Silverman [7].Suppose that f1 and f2 ∈ Ht,α(β), with f1(z) = z −∞∑n=2|an|zn andf2(z) = z−∞∑n=2|cn|zn with respect to g1 and g2 ∈ T , with g1 = z−∞∑n=2|bn|znand g2 = z −∞∑n=2|dn|zn.We will show, for 0 ≤ λ ≤ 1, thatλf1(z) + (1− λ)f2(z) = z −∞∑n=2γn(λ)zn ∈ Ht,α(β)with respect toλg1(z) + (1− λ)g2(z) = z −∞∑n=2δn(λ)zn ∈ Twhere γn(λ) = λ|an|+ (1− λ)|cn| and δn(λ) = λ|bn|+ (1− λ)|dn|.Since∞∑n=2{[tn+ (1− t)n2]γn(λ)− β [(1− α)n+ α] δn(λ)}=∞∑n=2{[tn+ (1− t)n2][λ|an|+ (1− λ)|cn|]β [(1− α)n+ α] [λ|bn|+ (1− λ)|cn|]}= λ∞∑n=2{[tn+ (1− t)n2]|an| − β [(1− α)n+ α] |bn|}(1− λ)∞∑n=2{[tn(1− t)n2]|cn| − β [(1− α)n+ α] |dn|}≤ 1− β,then the result follows.Theorem 5. If f(z) ∈ Ht,α(β), then f(z) is convex in the disc |z| <r = r(t, α, β) wherer(t, α, β) = infn{t+ (1− t)nαβ + (1− αβ)n} 1n−1, n = 2, 3, . . .This result is sharp. The extremal function isf(z) = z − αβ + (1− αβ)nn2[t+ (1− t)n] zn.420 D. PashkoulevaP r o o f. It suffices to show that∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣ ≤ 1 for |z| < 1.First we note that∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣ =∣∣∣∣∣∣∣∣∣∣∞∑n=2n(n− 1)|an|zn−11−∞∑n=2n|an|zn−1∣∣∣∣∣∣∣∣∣∣≤∞∑n=2n(n− 1)|an||z|n−11−∞∑n=2n|an||z|n− 1.Thus∣∣∣∣zf ′′(z)f ′(z)∣∣∣∣ ≤ 1 whenever∞∑n=2n(n− 1)|an||z|n−1 ≤ 1.But in view of Remark A [3] we have∞∑n=2[tn+ (1− t)n2]|an| ≤ αβ + (1− αβ)nn(6)where we have used the fact that |bn| ≤ 1n[5].Hence, f(z) is convex ifn2|z|n−1 ≤ n2[t+ (1− t)n2]αβ + (1− αβ)nthat is|z| ≤{t+ (1− t)nαβ + (1− αβ)n} 1n−1.Thus the radius of convexity is given byr(t, α, β) = infn{t+ (1− t)nαβ + (1− αβ)n} 1n−1.Theorem 6. If f1(z) = z −∞∑n=2|an|zn and f2(z) = z−∞∑n=2|cn|zn are inHt,α(β), thenh(z) = z − 12∞∑n=2|an + cn|znis also in Ht,α(β).SOME NOTES ABOUT A CLASS OF UNIVALENT . . . 421P r o o f. The proof follows directly by appealing to Theorem 1, [3].In fact, if f1(z) and f2(z) belong to Ht,α(β), then from (6) we have∞∑n=2[tn+ (1− t)n2]|an| ≤ αβ + (1− αβ)nn(7)and∞∑n=2[tn+ (1− tn)2]|cn| ≤ αβ + (1− α)nn. (8)For h(z) to be in Ht,α(β), it is sufficient to show that12∞∑n=2[tn+ 1(1− t)n2]|an + cn| ≤ αβ + (1− αβ)nnwhich follows immediately on using (7) and (8).Theorem 7. Let f1(z) = z and fn(z) =αβ + (1− αβ)nn2 [t+ (1− t)n]zn. Thenf(z) ∈ Ht,α(β) if and only if it can be expressed in the formf(z) =∞∑n=1λnfn(z)where λn ≥ 0 and∞∑n=1λn = 1.P r o o f. Suppose that f(z) =∞∑n=1λnfn(z) = zλ1+[z − z − αβ4(2− t) t2]λ2+. . .+[z − αβ + (1− αβ)nn2 [t+ (1− t)n]zn]λn = z −∞∑n=2αβ + (1− αβ)nn2 [t+ (1− t)n]λnzn. Then∞∑n=2{n [t+ (1− t)n]λnαβ + (1− αβ)nn2 [t+ (1− t)n]}≤ αβ + (1− αβ)nn.Thus, by Theorem 1 and Remark A, f(z) ∈ Ht,α(β).Conversely, suppose f(z) ∈ Ht,α(β). By Theorem 2, we have|an| ≤ αβ + (1− αβ)nn2 [t+ (1− t)n] .Settingλn =n2 [t+ (1− t)n]αβ + (1− αβ)n |an|and422 D. Pashkoulevaλ1 = 1−∞∑n=2λn,we have thatf(z) =(λ1 +∞∑n=2λn)z −∞∑n=2αβ + (1− αβ)nn2 [t+ (1− t)n]λnzn= zλ1 +[z − 2− αβ4(2− t)z2]λ2 + . . .+[z − αβ + (1− αβ)n[t+ (1− t)n] zn]λn =∞∑n=1λnfn(z).This completes the proof of the theorem. The extreme points of Ht,α(β)are functions f1 and f2.References[1] V. P. Gupta and P. K. Jain, Certain classes of univalent functions withnegative coefficients. Bull. Austral. Math. Soc. 14 (1976), 409-416.[2] V. P. Gupta and P. K. Jain, Certain classes of univalent functions withnegative coefficients II. Bull. Austral. Math. Soc. 15 (1976), 467-473.[3] D. Pashkouleva, On a class of univalent functions with negative coeffi-cients. Math. and Education in Math. 39 (2010), 149-153.[4] A. Schild, On a class of functions Schlicht in the unit circle. Proc. Amer.Math. Soc. 5 (1954), 115-120.[5] H. Silverman, Univalent functions with negative coefficients. Proc. Amer.Math. Soc. 51 (1975), 109-116.[6] H. Silverman, Extremal properties for a family of α-convex functions.Houston J. Math. 12 (1986), 267-273.[7] H. Silverman and E. Silvia. Convex families of starlike functions. Hous-ton J. Math. 4 (1978), 263-268.Institute of Mathematics & InformaticsBulgarian Academy of Sciences”Acad. G. Bontchev” Str., Block 81113 Sofia – BULGARIAe-mail: donka zh vasileva@abv.bgReceived: GFTA, August 27-31, 2010

Some Notes about a Class of Univalent Functions with Negative Coefficients

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Some Notes about a Class of Univalent Functions with Negative Coefficients

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