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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 + (zup(z)/up(z)))2 − 1∣ < 1 or ∣((zu p(z))′/up(z))2 − 1∣ < 1. We also determine the condition on these parameters so that (4(p+(b+1)/2)/c)up'(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+(zhμ,p''(z)/hμ,q'(z))1+z. Moreover, the properties of Alexander transform of the function hμ,p(z) are discussed.

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