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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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