Nevanlinna characteristic and integral inequalities with maximal radial characteristic for meromorphic functions and for differences of subharmonic functions

Abstract

Let f f be a meromorphic function on the complex plane with Nevanlinna characteristic T ( r , f ) T(r,f) and maximal radial characteristic ln ⁡ M ( t , f ) \ln M(t,f) , where M ( t , f ) M(t,f) is the maximum of the modulus | f | |f| on circles centered at zero and of radius t t . A number of well-known and widely used results make it possible to estimate from above the integrals of ln ⁡ M ( t , f ) \ln M (t,f) over subsets E E on segments [ 0 , r ] [0,r] in terms of T ( r , f ) T(r,f) and the linear Lebesgue measure of E E . In the paper, such estimates are obtained for the Lebesgue–Stieltjes integrals of ln ⁡ M ( t , f ) \ln M(t,f) with respect to an increasing integration function m m , and the sets E E on which the function m m is not constant can have fractal nature. At the same time, it is possible to obtain nontrivial estimates in terms of the h h -content and h h -Hausdorff measure of the set E E , as well as their partial d d -dimensional power versions with d ∈ ( 0 , 1 ] d\in (0,1] . All preceding similar estimates known to the author correspond to the extreme case of d = 1 d=1 and an absolutely continuous integration function m m with density of class L p L^p for

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亚纯函数和次调和函数差的Nevanlinna特征和具有最大径向特征的积分不等式

设f是复平面上具有Nevanlinna特征T（r，f）T（r、f）和最大径向特征ln的亚纯函数⁡ M（t，f）\ln M（t、f），其中M（t（f）M（t）是以零为中心且半径为t t的圆上的模量|f|f|的最大值。许多众所周知和广泛使用的结果使得从上面估计ln的积分成为可能⁡ 根据t（r，f）t（r、f）和E的线性Lebesgue测度，分段[0，r][0，r]上子集E E上的M（t，f）\ln M（t、f）。本文对ln的Lebesgue–Stieltjes积分给出了这样的估计⁡ M（t，f）\ln M（t、f）关于递增积分函数M M，并且函数M M不恒定的集合E E可以具有分形性质。同时可以得到集合E E的h-内容和h-Hausdorff测度的非平凡估计，以及它们与d∈（0,1]d\in的部分d维幂形式（0,1]。作者已知的所有先前的类似估计都对应于d=1d=1的极端情况和密度为LpL^p的绝对连续积分函数m m

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