In this paper, we define both the upper and lower order of a sense-preserving harmonic mapping in D . We generalize to case some known results about holomorphic functions with positive show consequences function having finite order. addition, improve related result when there is equality distortion theorem for mappings Some examples are provided illustrate developed theory.

in the paper we establish some new results depending on thecomparative growth properties of composition of entire functionsusing relative $l^{ast }$-order (relative $l^{ast }$-lowerorder) as compared to their corresponding left and right factorswhere $lequiv lleft( rright) $ is a slowly changing function.

<p style='text-indent:20px;'>We construct Green's functions for second order parabolic operators of the form <inline-formula><tex-math id="M1">\begin{document}$ Pu = \partial_t u-{\rm div}({\mathbf A} \nabla u+ {\mathbf b}u)+ c} \cdot u+du $\end{document}</tex-math></inline-formula> in id="M2">\begin{document}$ (-\infty, \infty) \times \Omega $\end{document}<...