engleski

Testing Alzer's inequality for Mathieu series $S(r)$

Consider the Mathieu series $S(r)= \sum_{; ; n=1}; ; ^\infty 2n(n^2+r^2)^{; ; -2}; ; . We interpolate the Alzer's bilateral bounding inequalityin the following manner. We find intervals $I_1, I_2$ such that \begin{; ; align*}; ; \frac1{; ; r^2+\kappa_1}; ; \le 2\int_1^\infty \frac{; ; [\sqrt{; ; t}; ; ]^2}; ; {; ; (r^2+t)^3}; ; \, dt \le S(r), \qquad & r \in I_1\\ S(r) < 4\int_1^\infty \frac{; ; [\sqrt{; ; t}; ; ]}; ; {; ; (r^2+t)^3}; ; \, dt + 2\int_1^\infty \frac{; ; [\sqrt{; ; t}; ; ]^2}; ; {; ; (r^2+t)^3}; ; \, dt\, \le \frac1{; ; r^2+\kappa_2}; ; \, \qquad &r\in I_2. \end{; ; align*}; ; Here $\kappa_1=1/(2\zeta(3)), \kappa_2=1/6$.

Mathieu series; Alzer's bilateral inequality

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