tailieunhanh - Some radius problem for certain families of analytic functions

The aim of this paper is to give bounds of the radius of α convexity for certain families of analytic functions in the unit disc. The radius of α-convexity is generalization of the radius of convexity and the radius of starlikeness, and introduced by , and | Turk J Math 24 (2000) , 401 – 412. ¨ ITAK ˙ c TUB Some Radius Problem for Certain Families of Analytic Functions Ya¸sar Polato˘glu, Metin Bolcal Abstract The aim of this paper is to give bounds of the radius of α-convexity for certain families of analytic functions in the unit disc. The radius of α-convexity is generalization of the radius of convexity and the radius of starlikeness, and introduced by ; and [3,4] Key Words: Subordination principle, Carathedory functions, Janowski Starlike functions, Starlike functions of order β, The radius of Starlikeness, The radius of convexity, The radius of α-convexity. 1. Introduction Most radius problems lead to functions p(z) with positive real part, or some more restrictive condition Re p(z). Therefore, for our study we shall need the following definitions and the subordination principle. Subordination Principle: Let g(z) and f(z) be regular and analytic in D = {z | z | 0, then this function is Caratheodory functions. The class of these functions is denoted by P . If we use the subordination principle we have p(z) ∈ p if and only if p(z) ≺ 1+z . 1−z () The Class of Janowski Functions Let p(z) = 1 + b1 z + b2 z 2 + · · · be regular and analytic in D and satisfies the condition p(0) = 1, Re p(z) > 0, p(z) ≺ 1 + Az , 1 − Bz −1 12 , 0 ≤ β 0 (Caratheodory’s Class) 2) p(1 − 2β, −1) = p(β) is the set defined by Re p(z) > β 3) p(1, 0) = p(1) is the set defined by |p(z) − 1| 12 , 0 ≤ β 0 z f (z) −1 f(z) ∗ ∗ 6) S (β, −β) = S∗∗ (β) is the class defined by f 0 (z) 0. The inequality () was proved by [5]. On the other hand 1−A 1−A > 0 =⇒ µ = > 0 =⇒ −1 0 , 1 + A > 0 =⇒ 1 + A 1 +A 1−A 1−A = >0 Re µ = Re 1+A 1+A () From the relation () and () we have the inequality (). This shows that the lemma is true. 404 2 ˘ POLATOGLU, BOLCAL The radius of α-convexity of the class of S ∗ (A, B) is