### 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.

The purpose of the paper is to study the uniqueness problems of linear differential polynomials of entire functions sharing a small function and obtain some results which improve and generalize the related results due to J. T. Li and P. Li (2015). Basically we pay our attention to the condition $\lambda \left(f\right)\ne 1$ in Theorems 1.3, 1.4 from J. T. Li and P. Li (2015). Some examples have been exhibited to show that conditions used in the paper are sharp.

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.