Extremal problems for functions of positive real part with a constraint and applications (Q1095276)

Let P(a,b) denote the set of functions that are analytic in \(\Delta =\{z:| z| <1\}\) and satisfy Re p(z)\(>0\) for \(| z| <1\) and \(p(a)=b\) where \(0<a<1\) and \(b>1\). Let \(S^*(a,b)\) denote the set of functions analytic in \(\Delta\) and satisfying \(f(0)=0\), \(f'(0)=1\) and (zf'(z))/(f(z))\(\in P(a,b).\) The author finds sharp bounds on Re p(z), \(| p'(z)|\) and Re\(\{\) zp'(z)/p(z)\(\}\) where \(p\in P(a,b)\) and \(| z| =r\) \((0<r<1)\). For the family \(S^*(a,b)\), the author obtains sharp bounds on \(| f(z)|\) and \(| f'(z)|\), the radius of convexity and upper bounds on the coefficients \(a_ 2\), \(a_ 3\) of the power series for f. Similar results are shown for the family of functions defined by the conditions Re f'(z)\(>0\) for \(| z| <1\), \(f(0)=0\), \(f'(0)=1\) and \(f(a)=b\) where \(0<a<1\) and \(b>1\).