### A class of gap series with small growth in the unit disc.

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We prove a normality criterion for a family of meromorphic functions having multiple zeros which involves sharing of a non-zero value by the product of functions and their linear differential polynomials.

We give an upper estimate of Petrenko's deviation for a meromorphic function of finite lower order in terms of Valiron's defect and the number p(∞,f) of separated maximum modulus points of the function. We also present examples showing that this estimate is sharp.

We give a necessary and sufficient condition for an analytic function in ${H}^{1}$ to have real part in class $L$$logL$. This condition contains the classical one of Zygmund; other variants are also given.