| This paper mainly investigates the properties of logarithmically complete monotonic-ity, complete monotonicity, convexity, monotonicity and relevant inequalities for the gamma function, psi function and polygamma functions. It can be divided into three parts:(I)The theory of (logarithmically) complete monotonicity:This paper has generalized the coefficient of variable of some ratios of the gamma function, and obtained the logarithmically complete monotonicity properties of these functions. As to Gauss' hypergeometric series, we further generalize the existing results and obtain the sufficient and necessary conditions for logarithmically complete monotonicity of some relevant functions. In addition, logarithmically complete monotonicity of some relevant the psi functions and polygamma functions has been proved by means of the convolution theorem of Laplace transforms, and hence we get two double inequalities, whose bounds are best possible. Finally, we elaborate on the relation between the theory of Bernstein and logarithmically complete monotonicity properties, and also discuss strongly complete monotonicity of the relevant psi functions.(II) The theory of inequality:We have discussed the Gautschi inequalities in great detail and generalized the latest results. Several functions, which are logarithmically complete mono-tonic, have been constructed and thus the study on Gaustschi inequalities can be generalized to the properties of functions. At the same time, we generalize and sharpen some inequalities involving the gamma function.(Ⅲ)Other respects:We have discussed the Schur-convexity properties of ratios of the gamma function and the problem of characterizations of the gamma function. As to the polygamma functions, we particularly investigate its monotonicity and generalize the results of Alzer et al. Several important results have been obtained. Finally, by the study of strongly complete monotonicity, super-additive and star-shaped functions are discussed. |
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