Abstract
In this paper, we show that in order for a proper compact subset of plane to be convex, it is necessary and sufficient that inverse norm function be subharmonic.
1. Introduction
A function , where is a domain in , is said to be subharmonic if it is upper semicontinuous, not identically , and satisfies the sub mean value inequality: its average over the boundary of each ball contained in is greater than or equal to its value at the center.
Let be a nonempty closed subset of . By , we denote the complement of in and by the convex hull of . We define the distance function from by where denotes the Euclidean norm.
For a given in and a positive real number , denotes the sphere centre and radius and denotes the open ball of centre and radius .
Motzkin’s Theorem ([1], Theorem 7.8) states that a nonempty closed set in is convex if and only if every point in has a unique nearest point in . Armitage and Kuran ([3], Theorem 3) used this result to prove that is subharmonic in if and only if is convex. Parker [4] prove a local Motzkin-type theorem in order to obtain a local version of Armitage and Kuran’s result in the case where .
Theorem 1. Let be a nonempty proper closed subset of and let be a domain such that . Then, is subharmonic in if and only if is convex.
He also showed that Theorem 1 does not hold in higher dimensions (see counterexample in [4]).
Now, let be a compact subset of the plane and be the Lebesgue measure concentrated on , i.e., . Consider the multiplication by operator , i.e., for each in . It is easy to check that is normal. Let and put , so . Since is regular, then, . Now, define so and , that is, , where is the spectrum of . Let , then, there is an open set with and . Define
There is such that . If , then, . Therefore, a.e. and so . Define the operator on by ; then, we have a.e. Thus, is not in and so . Thus, for , we have (see [5], Proposition 3.9 p.198):
We prove that equality occurs if and only if is convex. To prove this point, let us recall some definitions. For a bounded linear operator on a Hilbert space , the numerical range is the image of the unit sphere of under the quadratic form associated with the operator. More precisely,
Thus, the numerical range of an operator, like the spectrum, is a subset of the complex plane whose geometrical properties should say something about the operator. One of the most fundamental properties of the numerical range is its convexity, stated by the famous Toeplitz-Hausdorff Theorem. The other important property of is that its closure contains the spectrum of the operator. is a connected set and for normal operator , Also, for , (see relation 4.6–7 of [6]). Therefore, if is the shift operator defined on , then, we have (Theorem 1.4–4. of [6]). If is convex, then,
It follows, by (4), (6), and Theorem 1, that
Theorem 2. If is a compact subset of the plane, then, is convex if and only if the function defined by is subharmonic in .
Corollary 3.
Corollary 4. If is a complex number such that is large enough, then,
Corollary 5. (Laplacian). in
Corollary 6. If , then,
Corollary 7. If , then,
Let be subharmonic in . Define functions by
Then, by the maximum principle, increases as increases, but the behavior of is often erratic. For instance, it can be for some values of . Nevertheless, if increases not too rapidly, then, there are senses in which the growth of is controlled by that of .
The order of a subharmonic function on is defined by
Littlewood (1908) proved the existence of constants such that if is subharmonic in with finite order , then,
Littlewood stated his result for functions of the form with entire. His technique still works, though, for general subharmonic . Let us return now to Littlewood’s inequality (14), and let denote also the largest possible such constant. Littlewood showed that for we have . He conjectured (was confirmed) that for , the correct value should be . An extremal function would be , defined for by . Note that for , is harmonic, so that its Riesz mass is supported on the negative real axis. Also, for fixed , is a symmetric decreasing function of . Hence, so that the ratio is constant when . For more details, see [7].
Remark 8. Let , , the convex hull of and for some . For the function defined in Theorem 2, we have and , and then,
Also, and hence, .
Data Availability
No data was generated or analyzed during the current study.
Conflicts of Interest
The author declares that they have no conflicts of interest.