A problem in the theory of univalent functions

Abstract

Let ℋ_α (β) denote the class of normalized functions f, analytic in the unit disc E, which satisfy the condition:

where α and β are pre-assigned real numbers. Al-Amiri and Reade, in 1975, showed that, for α ≤ 0 and also for α = 1, the functions in ℋ_α (0) are univalent in E. In the present article, among other things, we prove: (i) for 0 < α < 1, functions in ℋ_α (α) are univalent in E and (ii) for α ≥ 0, functions f in ℋ_α (1/2) satisfy Re f′ (z) > 1/2 in E.

Keywords:

• Univalent functions
• Differential subordination
• Gaussian hypergeometric function