Inequalities on an extended Bessel function

• Published in: Journal of inequalities and applications Vol. 2018; no. 1; pp. 66 - 22
• Main Authors: Ali, Rosihan M., Lee, See Keong, Mondal, Saiful R.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2018; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Bessel function; Bessel functions; Concavity; Convexity; Differential equations; Formulations; Generalized Bessel function; Inequalities; Log-convexity; Mathematics; Mathematics and Statistics; Monotonicity properties; Parameters; Turán-type inequality; Bessel function; 26D7; Monotonicity properties; 33C10; Turán-type inequality; Log-convexity; Generalized Bessel function; 26D15
• ISSN: 1025-5834; 1029-242X; 1029-242X
• DOI: 10.1186/s13660-018-1656-4

This paper studies an extended Bessel function of the form B b , p , c a ( x ) : = ∑ k = 0 ∞ ( − c ) k k ! Γ ( a k + p + b + 1 2 ) ( x 2 ) 2 k + p . Representation formulations for B b , p , c a are derived in terms of the parameters a , b , and  p . An important consequence is the derivation of an ( a + 1 ) -order differential equation satisfied by the function B b , p , c a . Interesting functional inequalities are established, particularly for the case a = 2 , and c = ± α 2 . Monotonicity properties of B b , p , c a 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 B b , p , c d a ( x ) : = ∑ k = 0 ∞ ( − c / 4 ) k Γ ( p + b + 1 2 ) Γ ( k + 1 ) Γ ( a k + p + b + 1 2 ) ( d ) k k ! x k , which leads to direct and reverse Turán-type inequalities.