In this paper we consider properties and power expressions of the functions <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$f:(-1,1)\rightarrow \mathbb{R }$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$f_L:(-1,1)\rightarrow \mathbb{R }$$</EquationSource> </InlineEquation>, defined by <Equation ID="Equa1"> <EquationSource Format="TEX">$$\begin{aligned} f(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma }{\sqrt{1-t^2}}\,\mathrm{d}t \quad \text{ and}\quad f_L(x;\gamma )=\frac{1}{\pi }\int \limits _{-1}^1 \frac{(1+xt)^\gamma \log (1+x t)}{\sqrt{1-t^2}}\,\mathrm{d}t, \end{aligned}$$</EquationSource> </Equation>respectively, where <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$\gamma $$</EquationSource> </InlineEquation> is a real parameter, as well as some properties of a two parametric real-valued function <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$D(\,\cdot \,;\alpha ,\beta ) :(-1,1) \rightarrow \mathbb{R }$$</EquationSource> </InlineEquation>, defined by <Equation ID="Equa2"> <EquationSource Format="TEX">$$\begin{aligned} D(x;\alpha ,\beta )= f(x;\beta )f(x;-\alpha -1)- f(x;-\alpha )f(x;\beta -1),\quad \alpha ,\beta \in \mathbb{R }. \end{aligned}$$</EquationSource> </Equation>The inequality of Turán type <Equation ID="Equa3"> <EquationSource Format="TEX">$$\begin{aligned} D(x;\alpha ,\beta )>0,\quad -1>x>1, \end{aligned}$$</EquationSource> </Equation>for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$$\alpha +\beta >0$$</EquationSource> </InlineEquation> is proved, as well as an opposite inequality if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$$\alpha +\beta >0$$</EquationSource> </InlineEquation>. Finally, for the partial derivatives of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$$D(x;\alpha ,\beta )$$</EquationSource> </InlineEquation> with respect to <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$$\alpha $$</EquationSource> </InlineEquation> or <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$$\beta $$</EquationSource> </InlineEquation>, respectively <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$$A(x;\alpha ,\beta )$$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$$B(x;\alpha ,\beta )$$</EquationSource> </InlineEquation>, for which <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$$A(x;\alpha ,\beta )=B(x;-\beta ,-\alpha )$$</EquationSource> </InlineEquation>, some results are obtained. We mention also that some results of this paper have been successfully applied in various problems in the theory of polynomial approximation and some “truncated” quadrature formulas of Gaussian type with an exponential weight on the real semiaxis, especially in a computation of Mhaskar–Rahmanov–Saff numbers. Copyright Springer Science+Business Media New York 2013
