By Rajagopalan Parvatham, Saminathan Ponnusamy, K. S. Padmanabhan
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Extra info for Proceedings of an International Conference on New Trends in Geometric Function Theory and Applications: in honour of Professor K.S. Padmanabhan
JB . Reg'(z) >Szmplzes Re g(z) > S,ze U whenever g E s• (S). Corollary 4. If -1 s S < 1 and f, g e A, then Re /'(z) + ~zf"(z) 1 g'(z)+ 2zg"(z) f'(z) >SimpliesR eg'(z) > S,ze U whenever g e sc (S). For S = 0 the result of Corollary 3 was obtained by K. Sakaguchi in [11 ]. , af& implies fER,. (8,. (a)) 23 where 8, 8,. - 0. - 0 using the same technique like in the proof of Theorem 1 we obtain KRn, a (8) is equivalent to f e P() z + n+2p (z) + zP'(z) n (n + 2) (1 a -a) -a -< h () 6 z 1-(1-28)z 1 +z a where p (z) = a+ (n + 2) (1- a) _ (z) +a (n + 1) n+2 P n+2 and z wn f (z))' nn f (z) - (z)- p Letting P = (n + 2)/a and y - n (n + 2)(1 - a) - a 2 and using Theorem A we obtain that P (z) H( ) w = .
Corollary. Iff e R [A, B), then so are~ (j) and fix (j). Proof: The operators «l»y (j) and O:r (/) given by (12) and (13) are just two examples that can be realized as convolutions under specified convex functions. Remark: A direct proof that «l»y preserves R [A, BJ was given by V. Kumar [14), using Jack's Lemma. Next we consider opertors that are convex combinations of expressions involving functions from the given classes. For f (z) = z + L, anzn analytic in U and t ~ 0, we define n=2 At (j) = (1- t) f (z) + tzf'(z), z e U.
Ix lz I - a f( r 1 _ Bp) (A -B) (It+ 2)/(4aB) ( 1 + Bp)(A- B) (It+ 2)/(4aB) t 1/a-1 dp 0 which leads to the left-side inequality forB t:. 0 and similarly forB = 0. a For the sharpness of the inequalities, consider f e MIt [A, Bl . _-~~---::-:---:-::-- p lla- 1 dt, B t:. 0, -1 a F(z) = ( 1 _ BdA- B) (It + 2)/(4aB) 0 r ~ J e kAt12a t 1/a-1 dt, 0 6 B =0. Theorem 12: (a- convexity). The radius of a- convexity for IE M: fA. B) is r (1, 0) as in Theorem 6. Proofi Now (1- a) z['C$) I ($['($))' (z) +a j'C$) =P ($) e P, [A,B] and Rep($)> Ofor lzl < r (1, 0) as in the proof of Theorem 6.