On an integral inequality for certain analytic functions (Q2711411)

Let \(A_n\) denote the class of functions of the form \(f(z) = z + \sum_{k=1}^\infty a_{k+n} z^{k+n}\) which are analytic in the unit disc \(U\). Petru T. Mocanu proved a theorem involving an integral inequality which was equivalent to: If \(f(z) \in A_1\) and \(|f'(z) + zf''(z) - 1|< M_1\) for all \(z \in U\) then \(|zf'(z)/f(z) -1|< 1\) for all \(z \in U\). He first proved this with \(M_1 = 1\) but later improved it to \(M_1 = 8/(2 + \sqrt {15})\). In this paper, the author extends Mocanu's result to functions in the class \(A_n\) with \(M_1\) replaced by \(M_n = (n+1)^4/[n(n+3) + \sqrt{(n+1)^6 - 4n}]\).