Suppose v is twice continuously differentiable and...

Suppose $v$ is twice continuously differentiable and subharmonic in $\mathbb{C}$. Prove that
$$
\lim _{r \rightarrow \infty} \frac{M_1(r)}{\log r}
$$
exists (possibly infinite), where $M_1(r)$ is the mean value of $v$ on $|z|=r$.

Key Concepts

Subharmonic Functions

A subharmonic function is one that satisfies a mean value inequality, namely that its value at any point is less than or equal to the average value over any surrounding circle or sphere. When the function is twice continuously differentiable, this property is equivalent to having a non-negative Laplacian. This concept is central in potential theory and complex analysis, as it generalizes the notion of harmonic functions and is often utilized to study growth and boundary behavior of functions.

Mean Value Property

The mean value property involves calculating the average of a function along circles (or spheres) and is a key tool in studying harmonic and subharmonic functions. For harmonic functions, the value at the center exactly equals the mean value around the circle, while for subharmonic functions, it is bounded above by the mean value. In analyzing asymptotic behavior, such mean values are used to assess the growth of the function at infinity, providing critical insight into its scaling relative to standard functions like the logarithm.

Riesz Representation Theorem

This theorem provides a structural decomposition for subharmonic functions by expressing them as the sum of a harmonic function and a potential term generated by a positive measure (the Riesz measure). This decomposition is fundamental in understanding and estimating the contribution to the function's growth from its non-harmonic part, particularly when investigating asymptotic limits.

Asymptotic Growth Analysis

Asymptotic growth analysis is the study of a function's behavior as the independent variable tends to infinity. In the context of subharmonic functions on the complex plane, examining the limit of the ratio of the mean value on a circle to a logarithmic term as the radius increases provides a measure of the function's average rate of growth. This limit can either be finite or infinite and gives a global picture of the function’s long-term behavior.

Laplace Operator and Smoothness

The requirement that a function be twice continuously differentiable ensures that the Laplacian is well-defined in the classical sense, which is essential for verifying subharmonicity via the condition that the Laplacian is non-negative. This smoothness facilitates the use of analytic techniques and integration over circular domains, which are key components in proofs involving mean value properties and asymptotic analysis.

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