For let , where . Denote by the class of analytic functions in the unit disk . For we write ( is subordinate to in ) if there exists with for , such that . Also, for we let be the unique solution of the integral equation .
The paper under review is a continuation of the work of S. Koumandos and S. Ruscheweyh [J. Approximation Theory 149, No. 1, 42–58 (2007; Zbl 1135.42001)], where its authors guessed that:
Conjecture. For the number is indeed maximal number such that for all and we have
In the present paper, the authors give a proof of this conjecture in case . To do this, they prove (based on the properties of completely monotonic functions) a sharp trigonometric inequality. Main result of paper has some applications concerning starlike functions, as more as a corollary, it gives an inequality concerning Gegenbauer polynomials.