On p - valent functions with negative and missing coefficients

Year 1995, Volume: 44 , 0 - 0, 01.01.1995

M. K Aouf

https://doi.org/10.1501/Commua1_0000000452

Abstract

Let Pfc (p, A, B, a) be the class of functions f (z) — :p7A |a p - |zP+n’k - 2, analytic and p-valent in the unit disc U = |z: | z j < 1 | and satisfying the condition
1, z e U,
zf,(zL - p
'f ( z )
[pB + (A -B ) (P -a )l - B zf'(z) f(z)
where - 1 < B < A < 1, -1 < B < 0 , and 0 < a < p.
In this paper we obtain coefficient estimate, distortion and closure theorems and the radius of convexity for the class Pk (p, A, B, a). We also obtain class preserving integral ope- rators of the form
F(z) f t ^ 1 f(t) dt, e > - p Zç O
for the class Pk (p, A, B, a). Conversely when F (z) £ Pk (p, A, B, a), radius of p-valence of f (z) has been determined.

Keywords

p - valent, coefficients, missing

References

• Communications, Series A1:Mathematics and Statistics