On a multiplicative distribution of functions in complex plane

In the paper there are two kinds of functional symmetry considered, i.e., (1)-power symmetry and (-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1)$$\end{document}-power symmetry, in a rich subfamily B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {B}$$\end{document} of the family of all complex valued functions on a symmetric set G⊂C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}\subset \mathbb {C}$$\end{document}. The first one defines the values f(-z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(-z)$$\end{document} as identical with f(z) and the other as the inverse of the f(z) in G\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {G}$$\end{document}. The announced notions allow a unique decomposition of functions f∈B\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathcal {B}$$\end{document} into a product of two factors f1,f-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f_{1},f_{-1}$$\end{document} having the (1)- and the (-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-1)$$\end{document}-power symmetry property, respectively. In the paper there are also given examples of such partitions f=f1·f-1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f=f_{1}\cdot f_{-1}$$\end{document}, for various f∈B,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f\in \mathcal {B},$$\end{document} a solution of the problem of invariance of the above functional symmetries with respect to some one- and two-argument function operations and two applications of the mentioned function decomposition.


Introduction
It is known that for every complex valued function f on a symmetric set G ⊂ C (−1G = G), there exists exactly one function f E from the class F E (G) of even functions and exactly one function f O from the class F O (G) of odd functions on G, such that In view of the uniqueness of this partition, the functions f E and f O are called the even and odd part of f , respectively. This partition has interesting applications, see for instance the proof of the Borsuk sandwich theorem 1.4.1. in the paper [1] and see also [5]. Of course, not Definition 1. The function f ∈ B will be called (1)-power symmetric and (−1)power symmetric, if it satisfies the following conditions respectively.
For j = 1, −1 the family of all (j)-power symmetric functions f ∈ B will be denoted by B j . Obviously, the constant function f = 1 belongs to both families B 1 ,B −1 and there does not exist any other such function. Let us show that the families B j , j = 1, −1 are non-trivial.
Example. 1. Let the symmetric set G be the horizontal strip Then the function f (z) = exp(z), z ∈ G, belongs to B 1 .
Indeed, the function f belongs to B, because f is continuous, f (0) = 1 and Re exp(z) = e x cos y > 0 for z ∈ G. On the other hand , z ∈ G.
if we replace the horizontal strip G by the plane C.
2. Let the symmetric set G be the vertical strip Then the function f (z) = cos (z) , z ∈ G belongs to B 1 . Of course f ∈ B, For the relation f ∈ B 1 it suffices to observe that cos (−z) = cos (z) , z ∈ G. Note that f / ∈ B 1 if we replace the set G by the plane C. 3. Similarly, the function f (z) = cosh (z) = e z +e −z 2 , reduced to the above horizontal strip G, belongs to the family B 1 .
4. Let us observe, that the functions f (z) = sin (z) and f (z) = sinh (z) reduced to a symmetric set do not belong to B 1 or B −1 , because f (0) = 1.
5. Let the symmetric set G be the open unit disc U = {z ∈ C : |z| < 1} . Then the function Indeed, the function f belongs to B, because f is continuous, From now, we will always take the unit disc U as the symmetric set G.
In the paper, for a precise definition of the mentioned decomposition of a function into the product of a (1)-power symmetric and a (−1)-power symmetric function, we use the following very useful properties of the continuous unique main branch of the square root (the symbol √ · in the whole paper means the above main branch of the square root).

Lemma 2.1. Let
For z, w ∈ π + and the above main branch of the square root the following equalities hold: Note that Lemma 2.1. is a direct consequence of the following result [6, Chap. 5] for the continuous unique main branch of the logarithm Proposition 1. For z, w ∈ π + and the above main branch of the logarithm, the following equalities hold: Now, we give two theorems about the invariance of (j)-power symmetry j = 1, −1 with respect some functional operations.
We start with the case of operations of one function argument. Of course if f ∈ B, then the power f 0 = 1 belongs to both families B 1 , B −1 .
On the other hand we have the following.
Theorem 2.1. Let n be an integer, f ∈ B, its power f n belongs to B and √ · be the main branch of the square root. Then the following implications are true: Proof. (i) We take an n ∈ Z\{0}, then for f ∈ B 1 and z ∈ U we have: This gives the thesis (i). Similarly, for n ∈ Z\{0}, f ∈ B −1 and z ∈ U we have This gives the thesis (ii) .
. Then from the definition of the (1)power symmetric function, we have for z ∈ U: The next theorem concerns the invariance of the (j)-power symmetry j = 1, −1 with respect to some functional operations of two function arguments.
Proof. (i) Let f, g ∈ B 1 . Using the definition of the (1)-power symmetric function we show (i) as follows: Thus f · g is (1)-power symmetric.
(ii) Let f, g ∈ B −1 , then using the definition of the (−1)-power symmetric function we show similarly: Hence f · g is (−1)-power symmetric. For the proof of parts (iii) and (iv) it suffices to save f g = f · 1 g and use the above proved theses (i), (ii) and Theorem 2.   Proof. Firstly, we show that the functions f 1 , f −1 are well defined. Indeed, since Re f (−z) > 0, for z ∈ U, we also have Re 1 and Then the functions are well defined, because √ · means the main branch of the square root.
Now we show that f 1 is (1)-power symmetric and f −1 is (−1)-power symmetric. Let us note that f 1 (0) = 1 = f −1 (0) and f 1 , f −1 : U → π + . Obviously the functions f 1 and f −1 are continuous, because f is continuous, the product and quotient of continuous functions are continuous, the main branch of the square root is a continuous. From the form of f 1 , we have for z ∈ U: Thus f 1 is (1)-power symmetric. From the definition of f −1 and Lemma 2.1, we show that f −1 is (−1)-power symmetric. This follows from the following equalities for z ∈ U: This completes the proof of (i).
Next we show (ii). In view of thesis (i) and Lemma 2.1, we can write for z ∈ U the product f 1 · f −1 of the functions f 1 , f −1 , step by step: Now we will prove (iii), i.e., that the decomposition Then Thus, using the fact that g ∈ B 1 , h ∈ B −1 and Lemma 2.1, we get for z ∈ U : Similarly for z ∈ U: .
Using once again Lemma 2.1 and the fact that g ∈ B 1 , h ∈ B −1 , we get for z ∈ U : This gives (iii) and the proof is complete.
By the uniqueness of the above decomposition, we will call the function Putting −z in place of z, we obtain: Thus f is a (−1)-power symmetric function. Hence, by the uniqueness of the decomposition of the function f in Theorem 2.3., we have f 1 (z) = 1, and f −1 (z) = f (z), for z ∈ U .