In the last half-century, the study of Turán’s inequalities has been of great interest to pure and applied scientists. In particular, showing that some special functions and orthogonal polynomials satisfy this class of inequalities has been an active field of research.
It is pointed out by the authors that Turán’s inequalities related to special functions was successfully used to provide solutions to problems arising in information theory, economic theory and biophysics.
The paper deals with the deduction of sharp Turán inequalities for parabolic cylinder functions and Tricomi confluent hypergeometric functions. Also, monotonicity results concerning Turán determinants are treated.
At first, let the parabolic cylinder function \(U(a,.)\), also denoted \(D_{-a-1/2}\) following Whittaker’s notation, be defined as:
\[ U(a,x)= \frac{1}{2^{\eta} \sqrt{\pi}} \left[\cos(\eta \pi) \Gamma\left(\frac{1}{2} - \eta\right) y_{1}(a,x) - \sqrt{2} \sin(\eta \pi)\Gamma\left(1-\eta\right) y_{2}(a,x) \right], \]
where \(\eta=a/2+1/4\),
\[ y_{1}(a,x)=\exp\left(-\frac{x^2}{4}\right)\Phi\left(\frac{a}{2}+\frac{1}{4},\frac{1}{2},\frac{x^2}{2}\right), \] and
\[ y_{2}(a,x)=x\exp\left(-\frac{x^2}{4}\right)\Phi\left(\frac{a}{2}+\frac{3}{4},\frac{3}{2},\frac{x^2}{2}\right). \]
Further \(\Phi(a,c,.)\) stands for the Kummer confluent hypergeometric function, also known as the confluent hypergeometric function of the first kind.
In order to establish and prove the Turán’s inequalities for the parabolic cylinder functions, both the recurrence relations and differential equation satisfied by \(U(a,.)\) are employed. An integral formula related to the ratio of parabolic cylinder functions is explored in the proof context. Finally, the proof takes advantage of some properties of a modified Bessel function of the second kind \(K_{a}\).
Subsequently, the Turán inequalities for Tricomi’s \(\psi\) function
\[ \psi(a,c,x)=\frac{\Gamma(1-c)}{\Gamma(a-c+1)} \Phi(a,c,x) + \frac{\Gamma(c-1)}{\Gamma(a)} x^{1-c} \Phi(a-c+1,2-c,x) \]
are taken into account. The proof follows from previously defined inequalities and some conversion formulas between Tricomi functions and modified Bessel functions of the second kind. Also, the confluent differential equation and an integral representation for the ratio of Tricomi functions are employed.
The paper also deals with Turán determinants of Tricomi confluent hypergeometric functions. In fact, some complete monotonicity results are given concerning the latter determinants. The proof of these results follows standard approaches of integral and series manipulation.
The results are interesting and the paper is organized and written well. The structure is clear and the mathematical development is rigorous. Finally, the paper is accessible to a wide readership, including early graduate students with basic to medium knowledge on special functions.