Certain integrals for classes of p-valent meromorphic functions

LEMMA 2.1. The function Hz) = z + £ a zn belongs to S{m, M) n=2 " if and only if there exists a function w{z) regular in D and satisfying w(0) = 0 , \w{z)\ < 1 for z in D , such that (2 h) z h ' { z ) 1+a>w

LEMMA 2.2. The function f is in Y (m, M) if and only if there exists a function w(z) regular and satisfying w(0) = 0 , \w{z)\ < 1 , for z in D such that https://doi.org/10.1017/S0004972700005062 Published online by Cambridge University Press

LEMMA 2 . 3 . If the function w(z) is regular for \z\ 5 r < l , w(0) = 0 and \w(z)\ = max | u ( 3 ) | , then \z\=r , k > A proof of Lemma 2.3, which is due to Jack, may be found in [43.

LEMMA 2.4. A function f belongs to T*(p) , 0 2 p < 1 , if and only if there exists a function w(z) regular and satisfying u(0) = 0 , \w{z)\ < 1 in D such that https://doi.org/10.1017/S0004972700005062 Published online by Cambridge University Press

THEOREM 3.1. Let a, 3 and y be real constants such that a > 0 , g > 0 and y - p{a+&) + 1 > -1 . If f € T*(p) and g € T (m, M) , {m, M) € E* = {(m, M) : |m-p| < W 5 m*} where m* = min{m, (m-p)+(ap(l-p)/2g(y-pg-app+2))} , then I/a (3.1) F(z) = belongs to F*(p) . In (3.1) all powers are principal ones.

Theorem 3.1: (i) for a = 1 , 3 = 0 , y = 0 and p = 0 , a result of Bajpai [I]; (ii) for a = 3 = 1 , a result of Dwivedi [2]; (iii) for a = 3 , a result of Dwivedi, Bhargava and Shukla [3].

THEOREM 3.2. Let 3 and y be real constants such that 3 2 0 and y > 0 . If f e T*(p) and g € Y {m, M) , (m, M) € £•** = {(m, W) : |m-p| < M < where m* = min-jm, (m-p) Then (3.7) F(z) = J0 aZ-so belongs to r*(p) . In (3.7) all powers are principal ones.

THEOREM 4.1. If f € T (m, M) and F(z) is defined by (U.I) 1 I/a a > 0 . F(z) = F € T (m, W) if y > max l-£), pa-l} In (U.l) all powers are principal ones.

THEOREM 4.2. If f € E (m, M) and F(z) is defined by tyf{t)dt =\+ ... . 0 zP Then F € E (m, M) if y > max{ (p(l+a)+b-x)/(X-b), p-X] . In (U.10) all powers are principal ones.