Lemniscate convexity of generalized Bessel functions.

Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and "normalized" Bessel function up(z) = up,b,c(z) satisfies the inequalities ∣(1 + (zu″p(z)/u′p(z)))2 − 1∣ < 1 or ∣((zup(z))′/up(z))2 − 1∣ < 1. We also determine the condition on these parameters so that − (4 (p + (b + 1) / 2) / c) u p ' (x) ≺ 1 + z . Relations between the parameters μ and p are obtained such that the normalized Lommel function of first kind hμ,p(z) satisfies the subordination 1 + (z h μ , p ' ' (z) / h μ , q ' (z)) ≺ 1 + z . Moreover, the properties of Alexander transform of the function hμ,p(z) are discussed. [ABSTRACT FROM AUTHOR]