On the Behavior of Gegenbauer Polynomials in the Complex Plane

It is well-known that the squared modulus of every function f from the Laguerre–Polya class ${mathcal{L}-mathcal{P}}$ of entire functions obeys a MacLaurin series representation $$|f(x+i y)|^2=sum_{k=0}^{infty} L_k(f;x),y^{2k}, quad x,yinmathbb{R}$$ , which reduces to a finite sum when f is a polynomial having only real zeros. The coefficients {L _k } are representable as non-linear differential operators acting on f, and by a classical result of Jensen L _k (f;x)?≥ 0 for ${fin mathcal{L}-mathcal{P}}$ and ${xin mathbb{R}}$ . Here, we prove a conjecture formulated by the first-named author in 2005, which states that for ${f=P_n^{(lambda)} }$ , the n-th Gegenbauer polynomial, the functions ${{L_k(f;x)}_{k=1}^{n}}$ are monotone decreasing on the negative semi-axis and monotone increasing on the positive semi-axis. This result pertains to certain polynomial inequalities in the spirit of the celebrated refinement of Markov’s inequality, found by R. J. Duffin and A. C. Schaeffer in 1941.