# Fekete-Szegö Problem for a New Class of Analytic Functions

Authors: Deepak Bansal
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

We consider the Fekete-Szegö problem with complex parameterμ for the class Rγτ(ϕ) of analytic functions.


---

## Body

## 1. Introduction and Preliminaries

Let𝒜 denote the class of functions of the form (1.1)f(z)=z+∑k=2∞akzk,
which are analytic in the open unit disk 𝕌={z:z∈ℂand|z|<1} and 𝒮 denote the subclass of 𝒜 that are univalent in 𝕌. A function f(z) in 𝒜 is said to be in class 𝒮* of starlike functions of order zero in 𝕌, if ℜ(zf′(z)/f(z))>0 for z∈𝕌. Let 𝒦 denote the class of all functions f∈𝒜 that are convex. Further, f is convex if and only if zf′(z) is star-like. A function f∈𝒜 is said to be close-to-convex with respect to a fixed star-like function g∈𝒮* if and only if ℜ(zf′(z)/g(z))>0 for z∈𝕌. Let 𝒞 denote of all such close-to-convex functions [1].Fekete and Szegö proved a noticeable result that the estimate(1.2)|a3-λa22|≦1+2exp⁡(-2λ1-λ)
holds for any normalized univalent function f(z) of the form (1.1) in the open unit disk 𝕌 and for 0≦λ≦1. This inequality is sharp for each λ (see [2]). The coefficient functional (1.3)ϕλ(f)=a3-λa22=16(f′′′(0)-3λ2[f′′(0)]2),
on normalized analytic functions f in the unit disk represents various geometric quantities, for example, when λ=1, ϕλ(f)=a3-a22, becomes Sf(0)/6, where Sf denote the Schwarzian derivative (f′′′/f′)′-(f′′/f′)2/2 of locally univalent functions f in 𝕌. In literature, there exists a large number of results about inequalities for ϕλ(f) corresponding to various subclasses of 𝒮. The problem of maximising the absolute value of the functional ϕλ(f) is called the Fekete-Szegö problem; see [2]. In [3], Koepf solved the Fekete-Szegö problem for close-to-convex functions and the largest real number λ for which ϕλ(f) is maximised by the Koebe function z/(1-z)2 is λ=1/3, and later in [4] (see also [5]), this result was generalized for functions that are close-to-convex of order β.Letϕ(z) be an analytic function with positive real part on 𝕌 with ϕ(0)=1, ϕ′(0)>0 which maps the unit disk 𝕌 onto a star-like region with respect to 1 which is symmetric with respect to the real axis. Let 𝒮*(ϕ) be the class of functions in f∈𝒮 for which (1.4)zf′(z)f(z)≺ϕ(z)(z∈U),
and 𝒞(ϕ) be the class of functions in f∈𝒮 for which (1.5)1+zf′′(z)f′(z)≺ϕ(z)(z∈U),
where ≺ denotes the subordination between analytic functions. These classes were introduced and studied by Ma and Minda [6]. They have obtained the Fekete-Szegö inequality for the functions in the class 𝒞(ϕ).Motivated by the classRλτ(β) in paper [7], we introduce the following class.Definition 1.1.
Let0≦γ≦1, τ∈ℂ∖{0}. A function f∈𝒜 is in th class Rγτ(ϕ), if
(1.6)1+1τ(f′(z)+γzf′′(z)-1)≺ϕ(z)(z∈U),
where ϕ(z) is defined the same as above.
If we set(1.7)ϕ(z)=1+Az1+Bz(-1≦B<A≦1;z∈U),
in (1.6), we get
(1.8)Rγτ(1+Az1+Bz)=Rγτ(A,B)={f∈A:|f′(z)+γzf′′(z)-1τ(A-B)-B(f′(z)+γzf′′(z)-1)|<1},
which is again a new class. We list few particular cases of this class discussed in the literature(1)
Rγτ(1-2β,-1)=Rγτ(β) for 0≦β<1,τ=ℂ∖{0} was discussed recently by Swaminathan [7].(2)
The classRγτ(1-2β,-1) for τ=eiηcosη, where -π/2<η<π/2 is considered in [8] (see also [9]).(3)
The classR1τ(0,-1) with τ=eiηcosη was considered in [10] with reference to the univalencey of partial sums.(4)
f∈Rγeiηcosη(1-2β,-1) whenever zf′(z)∈Pγτ(β), the class considered in [11].
For geometric aspects of these classes, see the corresponding references. The classRγτ(A,B) is new as the author Swaminathan [7] has introduced class Rγτ(β) which is subclass of the class Rγτ(A,B), in his recent paper. To prove our main result, we need the following lemma.Lemma 1.2 (see [12, 13]).
Ifp(z)=1+c1z+c2z2+c3z3+⋯(z∈𝕌) is a function with positive real part, then for any complex number  μ,
(1.9)|c2-μc12|≦2max⁡{1,|2μ-1|},
and the result is sharp for the functions given by
(1.10)p(z)=1+z21-z2,p(z)=1+z1-z(z∈U).


## 2. Fekete-Szegö Problem

Our main result is the following theorem.Theorem 2.1.
Letϕ(z)=1+B1z+B2z2+B3z3+⋯, where  ϕ(z)∈𝒜 with  ϕ′(0)>0. If  f(z) given by (1.1) belongs to  Rγτ(ϕ)(0≦γ≦1,  τ∈ℂ∖{0},z∈𝕌), then for any complex number  μ(2.1)|a3-μa22|≦B1|τ|3(1+2γ)max⁡{1,|B2B1-3τμB1(1+2γ)4(1+γ)2|}.
The result is sharp.Proof.
Iff(z)∈Rγτ(ϕ), then there exists a Schwarz function w(z) analytic in 𝕌 with w(0)=0 and |w(z)|<1 in 𝕌 such that
(2.2)1+1τ(f′(z)+γzf′′(z)-1)=ϕ(w(z))(z∈U).
Define the function p1(z) by
(2.3)p1(z)=1+w(z)1-w(z)=1+c1z+c2z2+⋯.
Since w(z) is a Schwarz function, we see that ℜp1(z)>0 and p1(0)=1. Define the function p(z) by
(2.4)p(z)=1+1τ(f′(z)+γzf′′(z)-1)=1+b1z+b2z2+⋯.
In view of (2.2), (2.3), (2.4), we have
(2.5)p(z)=ϕ(p1(z)-1p1(z)+1)=ϕ(c1z+c2z2+⋯2+c1z+c2z2+⋯)=ϕ(12c1z+12(c2-c122)z2+⋯)=1+B112c1z+B112(c2-c122)z2+B214c12z2+⋯.
Thus,
(2.6)b1=12B1c1;b2=12B1(c2-c122)+14B2c12.
From (2.4), we obtain
(2.7)a2=B1c1τ4(1+γ);a3=τ6(1+2γ)[B1(c2-c122)+12B2c12].
Therefore, we have
(2.8)a3-μa22=B1τ6(1+2γ)(c2-νc12),
where
(2.9)ν=12(1-B2B1+3τμB1(1+2γ)4(1+γ)2).
Our result now is followed by an application of Lemma 1.2. Also, by the application of Lemma 1.2 equality in (2.1) is obtained when
(2.10)p1(z)=1+z21-z2orp1(z)=1+z1-z
but
(2.11)p(z)=1+1τ(f′(z)+γzf′′(z)-1)=ϕ(p1(z)-1p1(z)+1).
Putting value of p1(z) we get the desired results.For classRγτ(A,B), (2.12)ϕ(z)=1+Az1+Bz=(1+Az)(1+Bz)-1(z∈U)=1+(A-B)z-(AB-B2)z2+⋯.
Thus, putting B1=A-B and B2=-B(A-B) in Theorem 2.1, we get the following corollary.Corollary 2.2.
Iff(z) given by (1.1) belongs to  Rγτ(A,B), then
(2.13)|a3-μa22|≦(A-B)|τ|3(1+2γ)max⁡{1,|B+3τμ(A-B)(1+2γ)4(1+γ)2|}.



---

*Source: 143096-2011-08-07.xml*