On an integral operator for convex univalent functions

LEMMA 2.1. Let a,m and M be real numbers such that 0 < a < 1 and \m-l\ < M < m . If (2.1) t = i-a + am and T = aM , then K(t,T) ^K(m,M) .

LEMMA 2.2 Let a,m and M be real numbers such that and \m-l\ < M < m . If (2.4) t = 1-a + am and T = -aM , then KC£,T) £ K(m,M) .

THEOREM 3.1. If f e K(m,M) t then the function F , defined by (3.1) F(z) = £ {f'(u)}a du , also belongs to K(m3M) , provided (3.2) - <f> <, a < 1 where <t> = (M-\m-l\)/M+\m-l\) . The result is sharp.

COROLLARY 3.1. If f e K , then the function F defined by (3.1) also belongs to K , provided 0 ^ a S 1 . The result is sharp with the 2 -2 extremal function f(z) = / (1-u) du . o Very recently, Pandey and Bhargava [5] have shown that if f e K(m,U)} then the function F defined by (3.1) also belongs to K(m,M) provided a is a complex number such that |a| < (l-b)/2 , where b is given by (3.6). We show below that their result is incorrect: Let us consider the function (3.7) f(z) = :Z (i-bur(a+b)/bdu 0

THEOREM 3.2. If f e K(m,M) , then the function F defined by (3.1) also belongs to K(m,M) provided a is a complex number satisfying (3.8) \a\ < 1 and (3.9) \a\< 1 where <f> = (M-\m-l\ )/(M+\m-l\ ) .

COROLLARY 3.2. If / e K(m,M) , then the function F defined by (3.1) also belongs to K(m,M) provided a is a complex number satisfying (3.10) ] cc j < (J> . The result is sharp in the sense that the region given by (3.10) cannot be extended into any larger disc centred at the origin on the a-plane.