The Functional Equation max { χ ( xy ) , χ ( xy − 1 ) } = χ ( x ) χ ( y ) on Groups and Related Results

: This research paper focuses on the investigation of the solutions χ : G → R of the maximum functional equation max { χ ( xy ) , χ ( xy − 1 ) } = χ ( x ) χ ( y ) , for every x , y ∈ G , where G is any group. We determine that if a group G is divisible by two and three, then every non-zero solution is necessarily strictly positive; by the work of Toborg, we can then conclude that the solutions are exactly the e | α | for an additive function α : G → R . Moreover, our investigation yields reliable solutions to a functional equation on any group G , instead of being divisible by two and three. We also prove the existence of normal subgroups Z χ and N χ of any group G that satisfy some properties, and any solution can be interpreted as a function on the abelian factor group G / N χ .


Introduction
By the maximum functional equation on the group (G, +), we mean the functional equation max{ χ(x + y), χ(x − y) } = χ(x) + χ(y), x, y ∈ G, where χ : G → R is an unknown function to be determined. The solutions of the functional Equation (1) for real-valued functions on an abelian group G have been given by Volkmann and Simon [1]; they demonstrated that χ satisfies Equation (1) if and only if there is an additive function α : G → R such that χ(g) = |α(g)| for any g ∈ G. It has been shown that the functional Equation (1) characterizes the functions of the form χ(g) = |α(g)|. Subsequently, an interesting application of their result and a new proof were obtained by Kochanek [2]. In [3], Tabor generalized the functional Equation (1) for real and complex values.
In [4], Fechner dealt with four functional inequalities, which were derived from the results of [1] concerning the functional Equation (1) and also properties of the inequalities compared with some classical functional inequalities, such as the inequality of the Jensen quasi-convexity and the inequality of subadditivity.
Moreover, by rewriting the functional Equation (1) in the form Gilanyi et al. [5] determined the stability results of the functional Equation (2) on a square-symmetric groupoid. As a consequence, they also analyzed the stability of the functional Equation (1) on an abelian group. In addition, in [6] a Pexider version of the functional Equation (1) has been investigated. Later, their results were generalized by Badora et al. [7], who demonstrated the stability of the functional equation on a certain class of groupoids.
In [8], Jarczyck and Volkmann investigated the stability of Equation (1) on an abelian group and also presented a characterization of the functional Equation (1), which is formulated in the following theorem as follows: Theorem 1 (see [8]). Let G be an abelian group. Then, a function χ : G → R satisfies Equation (1) if and only if min{ The most exhaustive study of the functional Equation (1) on groups has been accomplished by Volkmann [9] and Toborg [10]. Volkmann generalized the functional Equation (1) in the form of with the additional assumption that χ satisfies the Kannappan condition [11], that is, χ(xyu) = χ(xuy) for all u, x, y ∈ G, and showed that it characterizes the absolute value of additive functions (see the theorem below). Recently, Toborg [10] gave the characterization of such functions in terms of (3) without additional assumptions (χ fulfills Kannappan condition and G is an abelian group). Their main theorem is stated as follows: Theorem 2 (see [10] for the general case and [9] for the special case of an abelian group G). Let G be any group; then a function χ : G → R satisfies Equation (3) if and only if there exists an additive function α : G → R such that χ(g) = |α(g)| for any g ∈ G.
Additionally, the characterization of the generalized functional Equation (3) and stability results can be found in [12]. In [1], Simon and Volkmann investigated the functional equation for abelian groups. The Equations (1) and (4) have complex analogues: Let V be a complex vector space and assume that A = {α | α ∈ C, |α| = 1} is the unit circle in C, and for functions χ : V → R, consider the equations for every x, y ∈ V. In [13] it has been proven that the solutions of the functional Equation (5) are given by χ(x) = |η(x)|, x ∈ V where η : V → C is a linear functional, and by Przebieracz [14] it was proven that the non-identically vanishing solutions of Equation (6) are of the form χ(x) = e |η(x)| , x ∈ V where η : V → C is also a linear functional; and in [14], Przebieracz proved the general theorem about the superstability of Equation (6). Baron with Volkmann [13] considered a characterization of the absolute value of complex determinants, which was similar to the results of Volkmann [15], who analyzed the real case using the functional Equation (1). Readers are encouraged to refer to [14,16] and the references cited therein to obtain useful results regarding the functional Equation (4).
The functional Equation (4) in a generalized form exhibited in equation is the focus of this research. No additional assumptions (G is an abelian group) are required, and thus, we derive results about solutions of the functional Equation (7) that are valid for any group rather than only in the special case of an abelian group G.
Additionally, to determine the solutions of the functional Equation (7) on a group G divisible by two and three, our objective is to implement the generalized result of the functional Equation (3) provided by Toborg [10] concerning the characterization of the modulus of an additive function. We will also present the existence of a certain normal subgroup N χ of G when a non-zero solution χ is unitary. Finally, we will present some results whenever the solutions of Equation (7) are strictly positive on an arbitrary group G.

Main Results
Throughout this article, 1 is considered to be the identity element of a group G, and G is an arbitrary group and divisible by two and three. Lemma 1. Let G be any group and χ : G → R is a strictly positive solution of the functional Equation (7). Then there is an additive function α : G → R such that χ(x) = e |α(x)| for all x ∈ G.
Proof. Since χ is a strictly positive solution of the functional Equation (7), we can compute Let η(x) = log χ(x); then η satisfies Equation (8). By applying the main theorem of Toborg [10], for any x ∈ G, there exists an additive function α : G → R such that η(x) = |α(x)|. Hence, by utilizing (8), for any x, y ∈ G, we can conclude that if and only if there exists an additive function α : One of the main theorems of this article is stated as follows: Theorem 3. Assume that each element of a group G is divisible by two and three; then a function χ : G → R fulfills the functional Equation (7) if and only if χ vanishes identically on a group G or there exists an additive function α : G → R such that χ(x) = e |α(x)| for any x ∈ G.
Proof. According to Lemma 1, we only have to prove that a nonzero solution χ is strictly positive.
The following corollary is a direct consequence of Theorem 3, so we omit the proof.

Corollary 1.
Assume that χ is a strictly positive solution of the functional Equation (7) on an arbitrary group G; then following hold: (3) For an infinitely divisible group G, χ(x q ) = qχ(x) for any rational number q > 0. Example 1. Assume that G = Z 6n−1 or G = Z 6n+1 for n ∈ N, then it can be perceived that each element of group G is divisible by two and three. Accordingly, by employing Theorem 3, we can obtain that χ(x) > 0 for every x ∈ G, and therefore, a nonzero function χ : G → R satisfies Equation (7) if and only if there exists an additive function α : G → R such that χ(x) = e |α(x)| for any x ∈ G. Example 2. Let G = Z 2 ; then each element of group G is divisible by three. Assume that On the other hand, let G = Z 3 ; then each element of G is divisible by two. Let χ(x) = 1 if x = 0 and χ(x) = −1 if x = 0; then χ also satisfies Equation (7) but χ(x) = e |α(x)| does not hold for all x ∈ G.
Let us consider G to be any group instead of divisible by two and three. Functional Equation (7) inspires us to investigate results about its non-zero solutions and numerous other results valid on any group G. Then we derive the following lemma that will be used a couple of times during proofs: Lemma 2. Let χ be a non-zero solution of the functional Equation (7) on any group G; then we can derive the following: (1) χ(1) = 1; Proof. (1). Since χ(x)χ(1) = max{ χ(x), χ(x) } and χ is non-zero, so, we have χ(1) = 1.
Moreover, we prove the following theorem by utilizing Lemma 2 to characterize the solutions χ of the functional Equation (7), without using additional assumptions about divisibility by two and three. xy −1 ∈ N χ or xy ∈ N χ for every x, y ∈ G and x, y / ∈ N χ ; or χ(x) = e |α(x)| for any x ∈ G and for some additive function α : G → R.

Corollary 2.
Let χ be a non-zero solution of the functional Equation (7) on any group G; then the commutator subgroup G is a normal subgroup of N χ .
Proof. Let χ be a non-zero solution of Equation (7). Then from Theorem 4, in the first case, assume that χ(x) > 0 for every x ∈ G; then there exists an additive function α : G → R such that χ(x) = e |α(x)| for every x ∈ G. Therefore, using the fact that α([x, y]) = 0, it can be seen that χ([x, y]) = 1 for every x, y ∈ G. In the second case, there exists a normal subgroup N χ of G such that χ(x) = 1 for all x ∈ N χ that also satisfies the condition (19); then the simple computation gives that which implies that χ([x, y]) = 1; therefore, in either case the commutator subgroup G is a normal subgroup of N χ . Definition 1. Let G be an arbitrary group. We say a function χ : G → R satisfies the Kannappan condition [11] if χ(xyu) = χ(xuy) for all u, x, y ∈ G. (7) on an arbitrary group G satisfies the Kannappan condition.

Corollary 3. Any solution of the functional Equation
Proof. The proof depends on three different cases, but in each case, the proof is quite obvious: Case 1: If χ = 0, then the result is trivial. Case 2: Let χ be strictly positive. The proof is quite simple because the additive function α(x) = log χ(x) for all x ∈ G.

Remark 2.
(1). It can be seen that Theorem 3 can also be concluded quite easily from Theorem 4, because if G is divisible by two and three, there is no nontrivial subgroup N χ with the required condition (19).
Proof. According to Corollary 3, the function χ satisfies the Kannappan condition; therefore, we have χ(x 2 ) = χ(xxyy −1 ) = χ(xyxy −1 ). Consequently, by Kannappan condition, the proof of proposition is a result of the following computation: We can get the following proposition, which is a direct consequence of Proposition 1, so we omit the proof.
The following theorem yields the information about the existence of a normal subgroup of G and will be important to compute more useful results.

Proposition 3.
Assume that χ is a non-zero solution of the functional Equation (7) on an arbitrary group G; then G N χ Z χ G.
Proof. The proof follows directly from Theorems 4 and 5 and Corollary 2.

Proposition 4.
If χ is a strictly positive solution of Equation (7) on an arbitrary group G, then Z χ = N χ .
In [10], Toborg has to be taken to prove that the solutions of the functional Equation (3) are of the form |α| for the additive function α : G → R, and that in the notation of the theorem, the relation ∼ R can be characterized as x ∼ R y if and only if α(x) and α(y) have the same sign (with the convention that 0 has both signs). The following propositions are obvious consequences of Theorem 3 and what Toborg used to prove her main result. The results of the following propositions are quite obvious and can be easily verified, so we overlook the proof.

Proposition 5.
Let χ be a strictly positive solution of the functional Equation (7) on an arbitrary group G; then relation ∼ R on group G satisfies the following: (1) ∼ R is reflexive; (2) ∼ R is symmetric; (3) Either x ∼ R y or x ∼ R y −1 ; (4) x ∼ R y if and only if x −1 ∼ R y −1 .

Proposition 6.
Let χ be a strictly positive solution of the functional Equation (7) on an arbitrary group G. If χ(z) = 1 for some z ∈ G, then the following results hold: (1) z ∼ R x for any x ∈ G; (2) If x ∼ R y then x ∼ R yz.
Proof. Since G is a strictly positive solution of Equation (7), there exists an additive function α : G → R such that χ(x) = e |α(x)| for any x ∈ G; therefore, the results become quite obvious because α(z) = 0.

Proposition 7.
Let χ be a strictly positive solution of Equation (7) on an arbitrary group G; then any solution χ : N χ → R can be interpreted as a function on the abelian factor group G/N χ .