On Some Monotonicity And Inequalities For The Psi And Polygamma Functions

To be the best of our knowledge,psi and polygamma functions have extensive applications in various fields.For example,they play an important role in the theory of special functions,inequality theory and statistics and are useful tools to study classical functions and constants.Moreover,a lot of graceful results have been obtained.Based on the previous research results,the main aim of this paper is to study the completely monotonic functions and inequalities involving the psi and polygamma functions,hence we improve and generalize some known results.Making use of the convolution theorem for Laplace transforms,the Bernstein's theorem and the convex function,we obtain some more mean-value inequalities for the psi and polygamma functions,study several completely monotonic functions for the polygamma functions,obtain some inequalities for ratios of gamma functions and differences of polygamma functions,and establish some new interesting results and generalise known results.The main results of this paper are as follows:(1)We obtain some mean-value inequalities for the psi and polygamma functions,and also give star-shaped functions which yield new inequalities.To be exactly,we determine necessary and sufficient conditions for the monotonicity and convexity of the functionF(x;?,?)= ln(exp(??(x + ?))?n(x))-ln(n-1)!,x > max(0,-?),where ?,? ? R and ?n(x)=(-1)n-1?(n)(x).Consequently,by using these properties,we provide upper and lower bounds involving the psi function for(?).In addition,we find sufficient and necessary conditions for(?),which is opposite to the result of(Alzer 2001),where M_n~[t] denotes power means.Finally,star-shaped functions are obtained.(2)By employing the convolution theorem for Laplace transforms,the Bernstein's theorem and the analytic function theory,we establish necessary and sufficient conditions for functionsL(x;?,?,?)= ?m+n(x)-??m(x + ?)?n(x + ?),?,?,? ? R,and?(x;?,s,t)= ??m+n(x)-?n(x)?m(x),?,s,t ? R,r = min(s,t),to be completely monotonic,whereThese two results generalize(Qi and Guo 2010: Theorem 3)and(Qi and Guo 2009b:Theorem 1),as the parameters increase and the order of the polygamma function becomes higher and more free.By the aid of these completely monotonic functions,we refine,to be extend,certain inequality,which is proved firstly by Batir and immediately give upper and lower bounds for the ratio [?n(x)?m(x)] /?m+n(x).Next,we offer the monotonicity of certain functions defined in terms of the the divided difference of psi and polygamma functions and the ratio of gamma functions,and therefore generalise the result of(Elezovi ?c et al.2000).Finally,taking into consideration these above results,we obtain a new inequality with the ratio of gamma functions.