Fekete–Szegö problem for subclasses of analytic functions defined by Komatu integral operator

Using the Komatu integral operator, new subclasses of analytic functions are introduced. For these classes, several Fekete–Szegö type coefficient inequalities are derived.


Introduction and definitions
Let A denote the class of functions of the form f (z) = z + a 2 z 2 + a 3 z 3 + · · · (1. 1) which are analytic in the unit disk U = {z ∈ C : |z| < 1}.
Also let S denote the subclass of A consisting of univalent functions in U.
Fekete and Szegö proved a noticeable result that the estimate holds for f ∈ S and for 0 ≤ λ ≤ 1. This inequality is sharp for each λ (see [8]). The coefficient functional on f ∈ A represents various geometric quantities as well as in the sense that this behaves well with respect to the rotation, namely In fact, other than the simplest case when we have several important ones. For example, Moreover, the first two non-trivial coefficients of the n-th root transform of f with the power series (1.1), are written by Thus, it is quite natural to ask about inequalities for φ λ corresponding to subclasses of S. This is called Fekete-Szegö problem. Actually, many authors have considered this problem for typical classes of univalent functions (see, for instance [1][2][3][4][5][6]8,[11][12][13]15,16]).
Using the operator L δ a , we now introduce the following classes: In particular, we have starlike and convex function classes,

respectively.
We denote by P a class of the analytic functions in U with p(0) = 1 and Re{ p(z)} > 0.
We shall require the following lemmas.
Then for any complex number ν and the result is sharp for the functions given by Lemma 1.5 [7] Let p ∈ P with p(z) = 1 + c 1 z + c 2 z 2 + · · ·. Then

2)
and Then by (1.3), we can write By the definition of the class S a,δ (b), there exists p ∈ P such that which implies the equality Equating the coefficients of both sides, we have so that, on account of (2.3) Taking into account (2.5) and Lemma 1.3, we obtain and Lemma 1.4 Moreover, by Lemma 1.3 Now, we consider functional |a 3 − μa 2 2 | for complex μ.

Moreover for each μ, there is a function in S a,δ (b) such that equality holds.
Proof Taking into account (2.5) we have Then, with the aid of Lemma 1.4, we obtain as asserted. An examination of the proof shows that equality is attained for the first case when c 1 = 0 and c 2 = 2 and the corresponding f ∈ S a,δ (b) is given by and likewise for the second case when c 1 = c 2 = 2 the corresponding f ∈ S a,δ (b) is given by respectively.
Taking δ = 0 and b = 1 in Theorem 2.2, we have Corollary 2.3 [12] If f ∈ S * , then for μ ∈ C we have Moreover for each μ, there is a function in S * such that equality holds.
We next consider the case when μ and b are real. Then we have: ,δ (b), then for μ ∈ R, we have Moreover for each μ, there is a function in S a,δ (b) such that equality holds. Proof By (2.7), we obtain In this case, by (2.12), Lemma 1.3 and Lemma 1.5 give Then, using the above calculations, we get Finally, if μ ≥ 1+b 2b ( a(a+2) (a+1) 2 ) δ , then we obtain Equality is attained for the second case on choosing c 1 = 0, c 2 = 2 in (2.9) and in (2.10) c 1 = c 2 = 2; c 1 = 2i, c 2 = −2 for the first and third case, respectively. Thus the proof is complete. Using the relation (1.6) , we easily obtain bounds of coefficients and a solution of the Fekete-Szegö problem in C a,δ (b). Moreover for each μ, there is a function in C a,δ (b) such that equality holds.
Taking δ = 0 and b = 1 in Theorem 2.6, we have Corollary 2.7 [12] If f ∈ C, then for μ ∈ C we have Moreover for each μ, there is a function in C such that equality holds.
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