Logarithmically absolutely monotone trigonometric functions

O. L. Vinogradov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 49, Tome 503 (2021), pp. 57-71

Résumé

We study absolute monotonicity and logarithmic absolute monotonicity for functions $$ f(z)=\dfrac{\cos{\alpha_1z}\cdot\ldots\cdot\cos{\alpha_Mz} \cdot\sin{\beta_1z}\cdot\ldots\cdot\sin{\beta_Nz}} {\cos{\alpha'_1z}\cdot\ldots\cdot\cos{\alpha'_{M'}z} \cdot\sin{\beta'_1z}\cdot\ldots\cdot\sin{\beta'_{N'}z}}z^{N'-N}. $$ Here $N,M,N',M'\in\Bbb Z_+$, $\alpha_j,\alpha_j',\beta_j,\beta_j'\geqslant 0$; if $\beta=0$, then the factor $\sin{\beta z}$ is replaced by $z$; if $N,M,N'$, or $M'$ equals zero, then the corresponding factors are absent. A criterion of logarithmic absolute monotonicity for $f$ is obtained. We give some applications of absolute monotonicity to sharp inequalities for derivatives and differences of trigonometric polynomials and entire functions of exponential type.