Research Projects

P. Turan discovered an inequality for the distribution of zeros of Legendre polynomials, known as the Turan inequality. G. Szego provided four elegant proofs for the Turan inequality, inspiring techniques for various polynomials and special functions. This led to a vast literature on Turan type inequalities for special functions, which are now widely used in spherical harmonics, potential theory, quantum mechanics, and other branches of applied mathematics and theoretical physics. The associated Legendre functions and the Ferrers functions are solutions of the associated Legendre differential equation, which can be reduced to the Legendre polynomials. The project aims to investigate functional inequalities and Turan type inequalities for associated Legendre and Ferrer functions. Turan type inequalities are closely connected to the Laguerre-Polya class (LP class) of real entire functions and the distribution of zeros of these functions. Analytical properties such as monotonicity, convexity, and log-concavity play a vital role in statistics and economics, motivating the project to investigate these properties. Associated Legendre and Ferrer functions have been studied in various perspectives, such as asymptotic expansions, orthogonality, and Fourier series in terms of associated Legendre functions. However, zeros of these functions have not been studied in detail. The next problem is to investigate the distribution and monotone properties of zeros of these functions.

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