By introducing the concept of X U -order functions, we study the growth of analytic functions defined by Laplace-Stieltjes transformations which converge on the right half-plane.
Kong, Sun, Huo and Xu investigated the growth of analytic functions with kinds of order defined by the Laplace-Stieltjes transforms (see [16 22]), and Shang, Gao, and Sun investigated the value distribution of such functions (see [23 26]).
The following lemma is very crucial in the study of the growth of analytic functions represented by Laplace-Stieltjes transforms convergent in the right half-plane which show the relation between M u ( σ, F ) and μ ( σ, F ) of such functions.
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This result generalizes the growth properties of analytic functions, harmonic functions, and superharmonic functions.
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