Convex Sets and Subharmonicity of the Inverse Norm Function

A function u : Ω⟶ 1⁄2−∞,∞Þ, where Ω is a domain in Rn, 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 Ω is greater than or equal to its value at the center. Let F be a nonempty closed subset of Rnðn ≥ 2Þ. By Fc, we denote the complement of F in Rn and by coðFÞ the convex hull of F. We define the distance function dF from F by


Introduction
A function u : Ω ⟶ ½−∞,∞Þ, where Ω is a domain in ℝ n , 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 Ω is greater than or equal to its value at the center.
Let F be a nonempty closed subset of ℝ n ðn ≥ 2Þ. By F c , we denote the complement of F in ℝ n and by coðFÞ the convex hull of F. We define the distance function d F from F by where k:k denotes the Euclidean norm. For a given x in ℝ n and a positive real number r, Sðx, rÞ denotes the sphere centre x and radius r and Bðx, rÞ denotes the open ball of centre x and radius r.
Motzkin's He also showed that Theorem 1 does not hold in higher dimensions (see counterexample in [4]). Now, let K be a compact subset of the plane and μ be the Lebesgue measure concentrated on K, i.e., μ = m 2 j K . Consider the multiplication by z operator A, i.e., ðAf ÞðzÞ = zf ðzÞ for each f in L 2 ðμÞ. It is easy to check that A is normal. Let s ∈ K and put U n = Bðs, 1/nÞ, so μðU n Þ ≠ 0. Since μ is regular, then, μðU n Þ < ∞. Now, define There is r > 0 such that Bðs, rÞ ⊂ U. If z ∈ U c , then, ð1/js − zjÞ < ð1/rÞ. Therefore, kψk ∞ ≤ ð1/rÞ a.e. and so ψ ∈ L ∞ ðμÞ. Define the operator T on L 2 ðμÞ by Tð f Þ = ψf ; then, we have ðs − AÞT = Tðs − AÞ = I a.e. Thus, s is not in σðAÞ and so σðAÞ = K. Thus, for z ∈ ℂ/K, we have (see [5], Proposition 3.9 p.198): We prove that equality occurs if and only if K is convex. To prove this point, let us recall some definitions. For a bounded linear operator T on a Hilbert space H, the numerical range WðTÞ is the image of the unit sphere of H under the quadratic form x ⟶ <Tx, x > 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 WðTÞ is that its closure contains the spectrum of the operator. WðTÞ is a connected set and for normal operator N, WðNÞ = coðσðNÞÞ: Also, for z ∉ WðTÞ, d WðTÞ ðzÞ ≤ kðz − TÞ −1 k −1 (see relation 4.6-7 of [6]). Therefore, if A is the shift operator defined on L 2 ðμÞ, then, we have WðAÞ = coðKÞ (Theorem 1.4-4. of [6]). If K is convex, then, It follows, by (4), (6), and Theorem 1, that Theorem 2. If K is a compact subset of the plane, then, K is convex if and only if the function u K : ℝ 2 ⟶ ½0,∞ defined by is subharmonic in ℂ.
Corollary 4. If z is a complex number such that jzj is large enough, then, Corollary 5. (Laplacian). Δu K ≥ 0 in K c : Corollary 7. If 1 ∉ K, then, Let u be subharmonic in ℂ. Define functions A, B : ½0, ∞Þ ⟶ ½−∞,∞Þ by Then, by the maximum principle, AðrÞ increases as r increases, but the behavior of AðrÞ is often erratic. For instance, it can be ∞ for some values of r. Nevertheless, if AðrÞ increases not too rapidly, then, there are senses in which the growth of A is controlled by that of B.
The order λ of a subharmonic function u on ℂ is defined by Littlewood (1908) proved the existence of constants Cð λÞ > −∞ such that if u is subharmonic in ℂ with finite order λ, then, Littlewood stated his result for functions of the form u = log j f j with f entire. His technique still works, though, for general subharmonic u. Let us return now to Littlewood's inequality (14), and let CðλÞ denote also the largest possible such constant. Littlewood showed that for 0 ≤ λ ≤ ð1/2Þ, we have CðλÞ ≥ cos ð2πλÞ. He conjectured (was confirmed) that for 0 ≤ λ < 1, the correct value should be CðλÞ = cos ð2πλÞ. An extremal function would be u λ , defined for θ ∈ ½−π, π by u λ ðre iθ Þ = r λ cos ðλθÞ. Note that for z ∈ ℂ − ð−∞,0Þ, u λ ðzÞ = Rez λ is harmonic, so that its Riesz mass is supported on the negative real axis. Also, for fixed r, u λ ðre iθ Þ is a symmetric decreasing function of θ. Hence, so that the ratio BðrÞ/AðrÞ is constant when u = u λ . For more details, see [7].

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.