ON A CLASS OF FUNCTIONS UNIFYING THE CLASSES OF PAATERO, ROBERTSON AND OTHERS

S. BHARGAVA ； S. NANJUNDA RAO

《International Journal of Mathematics and Mathematical Sciences》 1988卷 (1988/12) Pp. 251-258

https://doi.org/10.1155/S0161171288000304

摘要

We study a class M_k^λ(α, β, b, c) of analytic functions which unifies a number of classes studied previously by Paatero, Robertson, Pinchuk, Moulis, Mocanu and others. Thus our class includes convex and starlike functions of order β, spirallike functions of order β and functions for which zf' is spirallike of order β, functions of boundary rotation utmost kπ, α-convex functions etc. An integral representation of Paatero and a variational principle of Robertson for the class V_k of functions of bounded boundary rotation, yield some representation theorems and a variational principle for our class. A consequence of these basic theorems is a theorem for this class M_k^λ(α, β, b, c) which unifies some earlier results concerning the radii of convexity of functions in the classV_k^λ(β) of Moulis and those concerning the radii of starlikeness of functions in the classes U_k of Pinchuk and U_2(β) of Robertson etc. By applying an estimate of Moulis concerning functions in V_k^λ(0), we obtain an inequality in the class M_k^λ(α, β, b, c) which will contain an estimate for the Schwarzian derivative of functions in the class V_k^λ(β) and in particular the estimate of Moulis for the Schwarzian of functions in V_k^λ(0).

關鍵字

Convex functions ； starlike functions ； spirallike functions ； functions of bounded boundary rotation ； α-convex functions