Extremal problems for functions of positive real part and applications

Let B be the class of functions w(z) regular in |z|< I and satisfying w(0) = 0 , lw(z)I < 1 in Izl < I. We denote by P(A, B), -I ‚Äöv¢¬ß B < A ‚Äöv¢¬ß I, the class of functions p(z) = l+p l z+... regular in lz1 < I and such that p(z)=[1+Aw(z)]/[1+Bw(z)]for some w(z) ˜í¬µ B. This thesis is concerned with establishing bounds on z1=r<1 for functionals of the form Re{ap(z) + Up l (z)/p(z)} , a,˜í‚⧠real , where p(z) varies in P(A, B) or one of the following subclasses: Pk (A, B) = {p(z) = 1 +Pkzk+P2kz2k+‚ÄövÑvÆ ˜í¬µ P(A, B), k = 1,2,3,...} Pb (A, B) = {P(z) ˜í¬µ P(A2 B) ; P ' (0) = b(A-B) , 0 ‚Äöv¢¬ß b ‚Äöv¢¬ß 1} , P [a,b] = {P(z) ˜í¬µ P = P(1,-1) p(a)=b , 0 < a < I , b > 0} The bounds obtained are used to derive the distortion theorems, the covering theorems and the radii of convexity for the classes of regular or meromorphic starlike functions associated with P(A, B) or the above-mentioned subclasses. Furthermore, we obtain bounds for the functional Re{p(z)a-azp 1 (z)/p(z)}, 0 < a ‚Äöv¢¬ß 1 , p(z) E p , and establish the above theorems for the class of meromorphic strongly starlike functions of order a. The problem of minimising the functional Re{zp' (z)/p(z)} over P(A, B) is also examined for the case in which we may have A ‚Äöv¢‚Ä¢ I . This situation arises from the investigation of the starlikeness of functions f(z) normalised, regular in Iz| < I and satisfying |f(z)/[Af(z)+(1-A)g(z)]-y|