Lemniscate convexity of generalized Bessel functions

Abstract

AbstractSufficient conditions on associated parametersp, bandcare obtained so that the generalized and “normalized” Bessel functionup(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 . Relations between the parametersμandpare obtained such that the normalized Lommelfunction of first kindhμ,p(z)satisfies the subordination . Moreover, the properties of Alexander transform of the functionhμ,p(z) are discussed.