Investigation of some Pick functions and its integral representation
Abstract
In this research, we have proved that the function
which is analytic in the complex plane expect the negative real axis, where is a Gamma function, is a Pick function by proving that the imaginary part of the function is positive in the upper half complex plane expect the negative real axis. And we have obtained an integral representation of the function . From that, we have concluded that the derivative of the real function is a completely monotonic. Moreover, we have proved that some functions related to the function are Pick functions and we have found the integral representations for there.
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