On a Result of Levin and Ste？kin

The following inequality for 0<<1 and ≥0 originates from a study of Hardy, Littlewood, and Pólya: ∑∞=1∑((1/)∞=)≥∑∞=1. Levin and Stečkin proved the previous inequality with the best constant =(/(1？)) for 0<≤1/3. In this paper, we extend the result of Levin and Stečkin to 0<≤0.346.