Abstract
We prove that the function sigma(s) defined by beta(s)=6s2+12s+53s2(2s+3)-psi & PRIME;(s)2-sigma(s)2s5,s > 0, is strictly increasing with the sharp bounds 0 <sigma(s)< 49120, where beta(s) is Nielsen's beta function and psi & PRIME;(s) is the trigamma function. Furthermore, we prove that the two functions s?(-1)1+mu beta(s)-6s2+12s+53s2(2s+3)+psi & PRIME;(s)2+49 mu 240s5, mu=0,1 are completely monotonic for s > 0. As an application, double inequality for beta(s) involving psi & PRIME;(s) is obtained, which improve some recent results.