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Zbl 0924.30012
Darus, M.; Thomas, D.K.
$\alpha$-logarithmically convex functions.
(English)
[J] Indian J. Pure Appl. Math. 29, No.10, 1049-1059 (1998). ISSN 0019-5588; ISSN 0975-7465/e

Let $A$ denote the class of normalised analytic functions $f$ defined by $f(z) = z + a_2z^2 +\dots$ for $z\in D = \{z : | z| < 1\}$.\par For $\alpha\geq 0$ the authors introduce the class $M^\alpha$ of normalised analytic $\alpha$-logarithmically convex functions defined in the open unit disc $D$ by $$\text{Re} \left\{\left(1+ \frac{zf''(z)}{f'(z)}\right)^\alpha\left(\frac{zf'(z)}{f(z)}\right)^{1-\alpha}\right\} > 0.$$ For $f\in M^\alpha$, a best possible subordination theorem is obtained which implies that $M^\alpha$ forms a subset of the starlike functions $S^*$. Some extreme coefficient problems are also solved.\par This definition and some properties are inspired by the well known $\alpha$-convex functions, introduced in 1969 by P. T. Mocanu. The principal proofs of this paper are based on the so called admissible functions method'' defined and developed by P. T. Mocanu and S. S. Miller, method which used differential subordinations.
[Dorin Blezu (Sibiu)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions

Keywords: $\alpha$-logarithm; convex functions; extreme coefficient problem; Fekete-Szeg\H{o} functional

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