Radii of convexity for certain classes of univalent analytic functions

LEMMA 2. Let w e B. Then we have (2.2) Re ί ™Ά I < -_!_ReU(*) + -*--s-t 1(1 + sw(»))(l + ίw(2))l " ( s ί ) 2 I p ( ) where p(z) = (1 + tw(z))/(l + sw(z)), \z\ = r and

LEMMA 3. If p(z) = (1 + tw(z))/(l + sw(z)), w e B, then for each b e [0, 1] and s, t satisfying — l i * £ < s < ^ l , p(z) lies in the disc where A - (1 + br) z - str\b + r) 2. D

LEMMA 4. If p(z) = (1 + tw(z))J(l + z\ = r, 0 ^ r < 1, we 6r) 2 - s 2r 2(6 /or Ka + - r*)\p(z)\ (2.5) ^ where (2.5; a) (2.5; b)

THEOREM 1. Let f eRa(a, β), then f is convex in \z\<r0 where r0 is the smallest positive root of the equation 1 + iaβar + (Aaβ'a 2 - 2(1 + β - Zaβ))r z + 4β(2aβ - ί)ar 3 + (2/9 - l)(2αjβ - IV - 0 if Ra ^ R* and n - l{-aβ + Va{l - 2aβ + aβ 2)}/(l - if Ra ^ R* where R = 1 + 2aβar + {2aβ - l)r 2

THEOREM 2. Let geSα*(α, /9), then g is convex in \z\ < rQ where r0 is the smallest positive root of the equation 2/3(3α: - l)αr + (4a 2β 2a 2 + 8aβ - 2 - 4/9)r 2 - 2/5(1 + α - 4/9α 2)αr 3 + (1 - 2aβ) 2r< = 0 - [(5a - 1)/{(1 - a + 4/9α 2) + 4ατ/(l + β ^ R*, where

COROLLARY 2(a). Each geS%(a) maps | 2 | < r 0 onto a convex region where r0 is the smallest positive root of the equation - 2)ar + (4αV + 8a - 6)r 2 + (8a 2 - 2a - 2)αr 3 + (2a - 1)V = 0 if Ra ^ #* and r0 = [(5a - l)/{(4α 2 - a + 1) + AaV(a 9 - 3α