volume 6, issue 3, article 71, 2005.
Received 12 May, 2005;
accepted 24 May, 2005.
Communicated by:S. Saitoh
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Journal of Inequalities in Pure and Applied Mathematics
ON THE FEKETE-SZEGÖ PROBLEM FOR SOME SUBCLASSES OF ANALYTIC FUNCTIONS
T.N. SHANMUGAM AND S. SIVASUBRAMANIAN
Department of Mathematics College of Engineering
Anna University, Chennai-600 025 Tamilnadu, India.
EMail:shan@annauniv.edu Department of Mathematics Easwari Engineering College Ramapuram, Chennai-600 089 Tamilnadu, India.
EMail:sivasaisastha@rediffmail.com
c
2000Victoria University ISSN (electronic): 1443-5756 165-05
On the Fekete-Szegö Problem for Some Subclasses of
Analytic Functions
T.N. Shanmugam and S. Sivasubramanian
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Abstract
In this present investigation, the authors obtain Fekete-Szegö’s inequality for certain normalized analytic functions f(z) defined on the open unit disk for which (1−α)f(z)+αzfzf0(z)+αz2f00(z)0(z) (α ≥ 0)lies in a region starlike with respect to1and is symmetric with respect to the real axis. Also certain applications of the main result for a class of functions defined by convolution are given. As a special case of this result, Fekete-Szegö’s inequality for a class of functions defined through fractional derivatives is obtained. The Motivation of this paper is to give a generalization of the Fekete-Szegö inequalities obtained by Srivastava and Mishra .
2000 Mathematics Subject Classification:Primary 30C45
Key words: Analytic functions, Starlike functions, Subordination, Coefficient prob- lem, Fekete-Szegö inequality.
Contents
1 Introduction. . . 3 2 Fekete-Szegö Problem . . . 6 3 Applications to Functions Defined by Fractional Derivatives 10
References
On the Fekete-Szegö Problem for Some Subclasses of
Analytic Functions
T.N. Shanmugam and S. Sivasubramanian
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1. Introduction
LetAdenote the class of all analytic functionsf(z)of the form (1.1) f(z) =z+
∞
X
k=2
akzk (z ∈∆ :={z ∈C| |z|<1})
and S be the subclass of A consisting of univalent functions. Let φ(z) be an analytic function with positive real part on∆withφ(0) = 1, φ0(0) > 0which maps the unit disk∆onto a region starlike with respect to1which is symmetric with respect to the real axis. LetS∗(φ)be the class of functions inf ∈ S for which
zf0(z)
f(z) ≺φ(z), (z ∈∆) andC(φ)be the class of functions inf ∈ S for which
1 + zf00(z)
f0(z) ≺φ(z), (z ∈∆),
where ≺denotes the subordination between analytic functions. These classes were introduced and studied by Ma and Minda [10]. They have obtained the Fekete-Szegö inequality for the functions in the classC(φ). Sincef ∈ C(φ)if and only if zf0(z) ∈ S∗(φ), we get the Fekete-Szegö inequality for functions in the class S∗(φ). For a brief history of the Fekete-Szegö problem for the class of starlike, convex and close-to-convex functions, see the recent paper by Srivastava et al. [7].
In the present paper, we obtain the Fekete-Szegö inequality for functions in a more general classMα(φ)of functions which we define below. Also we give
On the Fekete-Szegö Problem for Some Subclasses of
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T.N. Shanmugam and S. Sivasubramanian
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applications of our results to certain functions defined through convolution (or the Hadamard product) and in particular we consider a class Mαλ(φ) of func- tions defined by fractional derivatives. The motivation of this paper is to give a generalization of the Fekete-Szegö inequalities of Srivastava and Mishra [6].
Definition 1.1. Letφ(z)be a univalent starlike function with respect to 1 which maps the unit disk ∆onto a region in the right half plane which is symmetric with respect to the real axis,φ(0) = 1andφ0(0) > 0. A functionf ∈ Ais in the classMα(φ)if
zf0(z) +αz2f00(z)
(1−α)f(z) +αzf0(z) ≺φ(z) (α≥0).
For fixedg ∈ A, we define the classMαg(φ)to be the class of functionsf ∈ A for which(f∗g)∈Mα(φ).
To prove our main result, we need the following:
Lemma 1.1. [10] Ifp1(z) = 1 +c1z+c2z2+· · · is an analytic function with positive real part in∆, then
|c2−vc21| ≤
−4v+ 2 ifv ≤0;
2 if0≤v ≤1;
4v−2 ifv ≥1.
Whenv <0orv >1, the equality holds if and only ifp1(z)is(1 +z)/(1−z) or one of its rotations. If0< v <1, then the equality holds if and only ifp1(z)
On the Fekete-Szegö Problem for Some Subclasses of
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is(1 +z2)/(1−z2)or one of its rotations. Ifv = 0, the equality holds if and only if
p1(z) = 1
2 +1 2λ
1 +z 1−z +
1 2 −1
2λ
1−z
1 +z (0≤λ≤1)
or one of its rotations. If v = 1, the equality holds if and only if p1 is the reciprocal of one of the functions such that the equality holds in the case of v = 0.
Also the above upper bound is sharp, and it can be improved as follows when 0< v < 1:
|c2−vc21|+v|c1|2 ≤2
0< v ≤ 1 2
and
|c2−vc21|+ (1−v)|c1|2 ≤2 1
2 < v ≤1
.
On the Fekete-Szegö Problem for Some Subclasses of
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T.N. Shanmugam and S. Sivasubramanian
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2. Fekete-Szegö Problem
Our main result is the following:
Theorem 2.1. Letφ(z) = 1 +B1z+B2z2+B3z3+· · ·. Iff(z)given by (1.1) belongs toMα(φ), then
|a3−µa22| ≤
B2
2(1+2α) − (1+α)µ 2B12+ 2(1+2α)(1+α)1 2B12 if µ≤σ1;
B1
2(1+2α) if σ1 ≤µ≤σ2;
−2(1+2α)B2 +(1+α)µ 2B12− 2(1+2α)(1+α)1 2B12 if µ≥σ2, where
σ1 := (1 +α)2(B2−B1) + (1 +α2)B12 2(1 + 2α)B12 , σ2 := (1 +α)2(B2+B1) + (1 +α2)B12
2(1 + 2α)B12 . The result is sharp.
Proof. Forf(z)∈Mα(φ), let
(2.1) p(z) := zf0(z) +αz2f00(z)
(1−α)f(z) +αzf0(z) = 1 +b1z+b2z2+· · ·. From (2.1), we obtain
(1 +α)a2 =b1 and (2 + 4α)a3 =b2+ (1 +α2)a22.
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Sinceφ(z)is univalent andp≺φ, the function p1(z) = 1 +φ−1(p(z))
1−φ−1(p(z)) = 1 +c1z+c2z2+· · · is analytic and has a positive real part in∆. Also we have
(2.2) p(z) =φ
p1(z)−1 p1(z) + 1
and from this equation (2.2), we obtain b1 = 1
2B1c1 and
b2 = 1
2B1(c2− 1
2c21) + 1 4B2c21. Therefore we have
(2.3) a3 −µa22 = B1
4(1 + 2α)
c2 −vc21 , where
v := 1 2
1−B2
B1 +(2µ−1) +α(4µ−α) (1 +α)2 B1
.
Our result now follows by an application of Lemma 1.1. To show that the bounds are sharp, we define the functionsKαφn (n= 2,3, . . .)by
z[Kαφn]0(z) +αz2[Kαφn]00(z)
(1−α)[Kαφn](z) +αz[Kαφn]0(z) =φ(zn−1), Kαφn(0) = 0 = [Kαφn]0(0)−1
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and the functionFαλ andGλα(0≤λ≤1)by z[Fαλ]0(z) +αz2[Fαλ]00(z)
(1−α)[Fαλ](z) +αz[Fαλ]0(z) =φ
z(z+λ) 1 +λz
, Fλ(0) = 0 = (Fλ)0(0)−1 and
z[Gλα]0(z) +αz2[Gλα]00(z)
(1−α)[Gλα](z) +αz[Gλα]0(z) =φ
−z(z+λ) 1 +λz
, Gλ(0) = 0 = (Gλ)0(0).
Clearly the functionsKαφn, Fαλ, Gλα ∈Mα(φ). Also we writeKαφ:=Kαφ2. Ifµ < σ1 orµ > σ2, then the equality holds if and only iff isKαφor one of its rotations. When σ1 < µ < σ2, the equality holds if and only iff isKαφ3 or one of its rotations. If µ= σ1 then the equality holds if and only if f isFαλ or one of its rotations. If µ= σ2 then the equality holds if and only iff isGλαor one of its rotations.
Remark 1. If σ1 ≤ µ ≤ σ2, then, in view of Lemma 1.1, Theorem2.1 can be improved. Letσ3be given by
σ3 := (1 +α)2B2+ (1 +α2)B12 2(1 + 2α)B12 . Ifσ1 ≤µ≤σ3, then
|a3−µa22|+ (1 +α)2 2(1 + 2α)B12
B1−B2+ (2µ−1) +α(4µ−α) (1 + 2α) B12
|a2|2
≤ B1
2(1 + 2α).
On the Fekete-Szegö Problem for Some Subclasses of
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Ifσ3 ≤µ≤σ2, then
|a3−µa22|+ (1 +α)2 2(1 + 2α)B12
B1+B2− (2µ−1) +α(4µ−α) (1 + 2α)2 B12
|a2|2
≤ B1 2(1 + 2α).
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3. Applications to Functions Defined by Fractional Derivatives
In order to introduce the classMαλ(φ), we need the following:
Definition 3.1 (see [3, 4]; see also [8, 9]). Let f(z) be analytic in a simply connected region of thez-plane containing the origin. The fractional derivative off of orderλis defined by
Dzλf(z) := 1 Γ(1−λ)
d dz
Z z 0
f(ζ)
(z−ζ)λdζ (0≤λ <1),
where the multiplicity of(z−ζ)λis removed by requiring thatlog(z−ζ)is real forz−ζ >0.
Using the above Definition3.1and its known extensions involving fractional derivatives and fractional integrals, Owa and Srivastava [3] introduced the op- eratorΩλ :A → Adefined by
(Ωλf)(z) = Γ(2−λ)zλDλzf(z), (λ 6= 2,3,4, . . .).
The class Mαλ(φ) consists of functions f ∈ A for which Ωλf ∈ Mα(φ).
Note that M00(φ) ≡ S∗(φ) andMαλ(φ)is the special case of the class Mαg(φ) when
(3.1) g(z) =z+
∞
X
n=2
Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) zn.
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Let
g(z) = z+
∞
X
n=2
gnzn (gn>0).
Since
f(z) =z+
∞
X
n=2
anzn ∈Mαg(φ) if and only if
(f∗g) = z+
∞
X
n=2
gnanzn∈Mα(φ),
we obtain the coefficient estimate for functions in the class Mαg(φ), from the corresponding estimate for functions in the classMα(φ). Applying Theorem2.1 for the function(f ∗g)(z) =z+g2a2z2+g3a3z3+· · ·, we get the following Theorem3.1after an obvious change of the parameterµ:
Theorem 3.1. Let the function φ(z) be given by φ(z) = 1 + B1z +B2z2 + B3z3+· · ·. Iff(z)given by (1.1) belongs toMαg(φ), then
|a3−µa22|
≤
1 g3
h B2
2(1+2α)− (1+α)µg32g22B21+ 2(1+2α)(1+α)1 2B21i
if µ≤σ1;
1 g3
h B1
2(1+2α)
i
if σ1 ≤µ≤σ2;
1 g3
h−2(1+2α)B2 + (1+α)µg32g22B12− 2(1+2α)(1+α)1 2B12i
if µ≥σ2,
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where
σ1 := g22 g3
(1 +α)2(B2−B1) + (1 +α2)B12 2(1 + 2α)B12
σ2 := g22 g3
(1 +α)2(B2+B1) + (1 +α2)B12 2(1 + 2α)B12 . The result is sharp.
Since
(Ωλf)(z) =z+
∞
X
n=2
Γ(n+ 1)Γ(2−λ) Γ(n+ 1−λ) anzn, we have
(3.2) g2 := Γ(3)Γ(2−λ)
Γ(3−λ) = 2 2−λ and
(3.3) g3 := Γ(4)Γ(2−λ)
Γ(4−λ) = 6
(2−λ)(3−λ).
Forg2 andg3 given by (3.2) and (3.3), Theorem3.1reduces to the following:
Theorem 3.2. Let the function φ(z) be given by φ(z) = 1 + B1z +B2z2 + B3z3+· · ·. Iff(z)given by (1.1) belongs toMαλ(φ), then
|a3−µa22| ≤
(2−λ)(3−λ)
6 γ if µ≤σ1;
(2−λ)(3−λ)
6 · 2(1+2α)B1 if σ1 ≤µ≤σ2;
(2−λ)(3−λ)
6 γ if µ≥σ2,
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where
γ := B2
2(1 + 2α) − 3(2−λ) 2(3−λ)
µ
(1 +α)2B12+ 1
2(1 + 2α)(1 +α)2B12,
σ1 := 2(3−λ)
3(2−λ).(1 +α)2(B2−B1) + (1 +α2)B12 2(1 + 2α)B12
σ2 := 2(3−λ)
3(2−λ).(1 +α)2(B2+B1) + (1 +α2)B12 2(1 + 2α)B12 . The result is sharp.
Remark 2. When α = 0, B1 = 8/π2 and B2 = 16/(3π2), the above Theo- rem 3.1 reduces to a recent result of Srivastava and Mishra[6, Theorem 8, p.
64] for a class of functions for which Ωλf(z) is a parabolic starlike function [2,5].
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References
[1] B.C. CARLSON AND D.B. SHAFFER, Starlike and prestarlike hyperge- ometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
[2] A.W. GOODMAN, Uniformly convex functions, Ann. Polon. Math., 56 (1991), 87–92.
[3] S. OWA AND H.M. SRIVASTAVA, Univalent and starlike generalized bypergeometric functions, Canad. J. Math., 39 (1987), 1057–1077.
[4] S. OWA, On the distortion theorems I, Kyungpook Math. J., 18 (1978), 53–58.
[5] F. RØNNING, Uniformly convex functions and a corresponding class of starlike functions, Proc. Amer. Math. Soc., 118 (1993), 189–196.
[6] H.M. SRIVASTAVA AND A.K. MISHRA, Applications of fractional cal- culus to parabolic starlike and uniformly convex functions, Computer Math. Appl., 39 (2000), 57–69.
[7] H.M. SRIVASTAVA, A.K. MISHRA ANDM.K. DAS, The Fekete-Szegö problem for a subclass of close-to-convex functions, Complex Variables, Theory Appl., 44 (2001), 145–163.
[8] H.M. SRIVASTAVAANDS. OWA, An application of the fractional deriva- tive, Math. Japon., 29 (1984), 383–389.
[9] H.M. SRIVASTAVA AND S. OWA, Univalent functions, Fractional Cal- culus, and their Applications, Halsted Press/John Wiley and Songs, Chich- ester/New York, (1989).
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[10] W. MA AND D. MINDA, A unified treatment of some special classes of univalent functions, in: Proceedings of the Conference on Complex Analy- sis, Z. Li, F. Ren, L. Yang, and S. Zhang(Eds.), Int. Press (1994), 157–169.