Inequalities on an extended Bessel function

Abstract

Abstract This paper studies an extended Bessel function of the form Bb,p,ca(x):=∑k=0∞(−c)kk!Γ(ak+p+b+12)(x2)2k+p. $$ {}_{a}\mathtt{B}_{b, p, c}(x):= \sum _{k=0}^{\infty }\frac{(-c)^{k}}{k! \Gamma { ( a k +p+\frac{b+1}{2} ) } } \biggl( \frac{x}{2} \biggr) ^{2k+p}. $$ Representation formulations for Bb,p,ca ${}_{a}\mathtt{B}_{b,p, c}$ are derived in terms of the parameters a, b, and p. An important consequence is the derivation of an (a+1) $(a+1)$-order differential equation satisfied by the function Bb,p,ca ${}_{a}\mathtt{B}_{b,p, c}$. Interesting functional inequalities are established, particularly for the case a=2 $a=2$, and c=±α2 $c=\pm \alpha^{2}$. Monotonicity properties of Bb,p,ca ${}_{a}\mathtt{B}_{b,p, c}$ are also studied for non-positive c. Log-concavity and log-convexity properties in terms of the parameters d and p are respectively investigated for the closely related function Bb,p,cda(x):=∑k=0∞(−c/4)kΓ(p+b+12)Γ(k+1)Γ(ak+p+b+12)(d)kk!xk, $$ {}_{a}\mathcal{B}^{d}_{b,p, c}(x): =\sum _{k=0}^{\infty }\frac{(-c/4)^{k} \Gamma { ( p+\frac{b+1}{2} ) }}{ \Gamma{ ( k+1 ) } \Gamma { ( ak+p+\frac{b+1}{2} ) }} \frac{(d)_{k}}{k!}x^{k}, $$ which leads to direct and reverse Turán-type inequalities.

Keywords

• Generalized Bessel function
• Bessel function
• Turán-type inequality
• Monotonicity properties
• Log-convexity