Distortion theorems for a special class of analytic functions

Abstract

Abstract.Sharp bounds are derived for the derivative of analytic functions of class .P,, defined by the condition |/>(z) -l/2a|<: l/2a, O^a^l, p(f>) = \. These results are improved for the class of functions with missing terms. Application is made to the class of functions with derivative G Px and the radius of convexity is determined for this class.

Radii of convexity for certain classes of univalent analytic functions

“…These were obtained by the authors in [9] and [5] and thus also include the results obtained in [1], [2], [13] etc.…”

“…Replacing (a, β) by (0,1), or by (0,1 -δ), 0 ^ δ < 1 or by (0, (25 -l)/2δ), 1/2 < δ ^ 1, or by ((1 -7)/l + 7, (1 + 7(/2), 0 < 7 ^ 1, or by ((1 -a + 27δ)/(l + δ), (1 + δ)/2), 0 ^ 7 < 1, 0 < δ ^ 1, we get the estimates for the radii of convexity for functions with 366 0. P. JUNEJA AND M. L. MOGRA fixed second coefficient of the classes introduced and studied by MacGregor [7], Shaffer [13], Goel [3], Caplinger and Causey [1] and the authors [6] respectively. 4* The class S α *(α, β).…”

On certain extremal problems for functions with positive real part

“…In the last section we shall derive the exact upper bounds of the functionals (Rep(z))/\p(z) + i tan a\ and \zp'(z)/(p(z) + i tan a)\ for p G P. Crude bounds, in particular of the second entity, have been used by several authors [1], [3], [7], but a sharp result has not so far been obtained. Let F(z) be analytic in A, F(0) =£ 0, and let t > 0.…”

“…License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use Lemma 1 (Suffridge [7,Theorem 7]). For /3 < 1 we have f E S% if and only if Spl * f * g E S| for all g E S%.…”

On Certain Extremal Problems for Functions with Positive Real Part

“…License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use Lemma 1 (Suffridge [7,Theorem 7]). For /3 < 1 we have f E S% if and only if Spl * f * g E S| for all g E S%.…”