Abstract
Recently, two classes of univalent functions S e * and K e were introduced and studied. A function f is in S e * if it is analytic in the unit disk, f ( 0 ) = f ′ ( 0 ) − 1 = 0 and z f ′ ( z ) f ( z ) ≺ e z . On the other hand, g ∈ K e if and only if z g ′ ∈ S e * . Both classes are symmetric, or invariant, under rotations. In this paper, we solve a few problems connected with the coefficients of functions in these classes. We find, among other things, the estimates of Hankel determinants: H 2 , 1 , H 2 , 2 , H 3 , 1 . All these estimates improve the known results. Moreover, almost all new bounds are sharp. The main idea used in the paper is based on expressing the discussed functionals depending on the fixed second coefficient of a function in a given class.
Highlights
Let A be the collection of functions of the form ∞ f (z) = z + ∑ an zn (1)n =2 which are analytic in the open unit disk ∆ = {z ∈ C : |z| < 1} and let S denote the subclass of A consisting of functions which are univalent in ∆.Since the early twentieth century many mathematicians have been interested in different problems involving the coefficients of functions f in a given subclass of A
It is worth adding that both estimates can be improved if a more precise inequality than | p3 − 12 p1 p2 | ≤ 2 for
2p − 1 p3, h i p ∈ 0, 43, h i p ∈ 43, 2. This gives for f ∈ Se∗
Summary
The sharp results for H3,1 are difficult to obtain It is worth citing the sharp bound | H3,1 | ≤ 4/135 for K and the non-sharp estimate | H3,1 | ≤ 8/9 for. Various problems, including distortion and growth theorems, radii problems, inclusion relations and coefficient estimates, were discussed there In their two following papers Zhang et al [6] and Shi et al [7] broadened the range of discussed coefficient problems. They found the coefficient estimates for an , n = 2, 3, 4, 5 and the bounds of the following functionals: a2 a3 − a4 , H2,1 , H2,2 and, as a consequence, H3,1. The new bounds of a2 a3 − a4 and H2,2 are sharp
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