On Turan-type inequalities for trigonometric polynomials of half-integer order

Abstract

Some exact inequalities of the Turan type are obtained in the paper for trigonometric polynomials $$$h(x)$$$ of half-interger order $$$n+\frac {1}{2}$$$, $$$n=1, 2, ...$$$, such that all $$$2n+1$$$ their zeros are real and located on a segment $$$[0;2\pi )$$$. Namely, the inequality that relates the norms in the space $$$C$$$ of the  trigonometric polynomials $$$h(x)$$$ of half-integer order $$$n+\frac {1}{2}$$$, $$$n=1, 2, ...$$$, and its second derivative $$$h''(x)$$$, $$$\|h''\|_c\ge \frac {2n+1}{4}\|h\|_c$$$, that is the inequalities that connect the norms in the space $$$L_2$$$ of the  trigonometric polynomials $$$h(x)$$$ of half-interger order $$$n+\frac {1}{2}$$$, $$$n=1, 2, ...$$$, and its first derivative $$$h'(x)$$$, that is $$$\|h'\|_{L_2}\ge \sqrt {\frac {2n+1}{8}}\|h\|_{L_2}$$$. The resulting inequalities cannot be improved. In proving the theorems, we use the method that was developed by V.F. Babenko and S.A. Pichugov for trigonometric polynomials, all of whose roots are real.