An Approximation Formula for Nielsen’s Beta Function Involving the Trigamma Function - Mansour Mahmoud; Hanan Almuashi (Mathematics, 2022)
We prove that the function <semantics>σ(s)</semantics> defined by <semantics>β(s)=6s2+12s+53s2(2s+3)−ψ′(s)2−σ(s)2s5,s>0</semantics>, is strictly increasing with the sharp bounds <semantics>0<σ(s)<49120</semantics>, where <semantics>β(s)</semantics> is Nielsen’s beta function and <semantics>ψ′(s)</semantics> is the trigamma function. Furthermore, we prove that the two functions <semantics>s↦(−1)1+μ<mfenced separators="" open="[" close="]">β(s)−6s2+12s+53s2(2s+3)+ψ′(s)2+49μ240s5</mfenced>, μ=0,1</semantics> are completely monotonic for <semantics>s>0</semantics>. As an application, double inequality for <semantics>β(s)</semantics> involving <semantics>ψ′(s)</semantics> is obtained, which improve some recent results.