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On a conjecture for trigonometric sums and starlike functions. II. (English) Zbl 1200.42002

For $\mu >0$ let ${s}_{n}^{\mu }\left(z\right)={\sum }_{k=0}^{n}{\left(\mu \right)}_{k}{z}^{k}/k!$, where ${\left(\mu \right)}_{k}=\mu \left(\mu +1\right)\cdots \left(\mu +k-1\right)$. Denote by $𝒜$ the class of analytic functions in the unit disk $𝔻=\left\{z\in ℂ:|z|<1\right\}$. For $f,g\in 𝒜$ we write $f\prec g$ ($f$ is subordinate to $g$ in $𝔻$) if there exists $w\in 𝒜$ with $|w\left(z\right)|\le z$ for $z\in 𝔻$, such that $f=g\circ w$. Also, for $\rho \in \left(0,1\right]$ we let ${\mu }^{*}\left(\rho \right)$ be the unique solution of the integral equation ${\int }_{0}^{\left(\rho +1\right)\pi }sin\left(t-\rho \pi \right){t}^{\mu -1}dt=0$.

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 $\rho \in \left(0,1\right]$ the number ${\mu }^{*}\left(\rho \right)$ is indeed maximal number $\mu \left(\rho \right)$ such that for all $n\in ℕ$ and $\mu \in \left(0,\mu \left(\rho \right)\right]$ we have

${\left(1-z\right)}^{\rho }{s}_{n}^{\mu }\left(z\right)\prec {\left(\frac{1+z}{1-z}\right)}^{\rho }·$

In the present paper, the authors give a proof of this conjecture in case $\rho =1/4$. 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.

##### MSC:
 42A05 Trigonometric polynomials, inequalities, extremal problems 30A10 Inequalities in the complex domain 26D05 Inequalities for trigonometric functions and polynomials 42A32 Trigonometric series of special types (positive coefficients, monotonic coefficients, etc.) 26D20 Analytical inequalities involving real functions