Selected Topics in the Classical Theory of Functions of a Complex Variable by Heins Maurice

Author:Heins, Maurice
Language: eng
Format: epub
Publisher: Dover Publications
Published: 2014-03-11T04:00:00+00:00

Thanks to these properties of [resp., u(z-m)] for z small. It is easily verified that in the first case the mean value of on C(0;r) for r small is just the mean value of u on C(φ(a);rm) and in the second case is the mean value of u on C(0;r-m). By Ex. 2, § 8, this chapter, property (d) follows.

EXERCISES

Most of the exercises that follow have as their theme the relation of convexity for functions of one real variable to subharmonicity. If we observe that the one-dimensional analogue of a harmonic function (= “solution” of Laplace’s equation) is a linear function (= solution of y″ = 0), we see that the subharmonic functions (functions having the harmonic-majorant property + upper semicontinuity) parallel the convex functions. Exercises 1 to 9, which follow, serve to elaborate this remark.

1. To be complete, we recall that a finite real-valued function f whose domain D is a connected subset of R is termed convex provided that whenever a,b(> a) ∈ D,

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