Properties of generalized $(p, q)$-elliptic integrals and generalized $(p, q)$-Hersch-Pfluger distortion function

1.   Introduction

In this paper, we primarily focus on investigating different properties of generalized $(p, q)$-elliptic integrals and the generalized $(p, q)$-Hersch-Pfluger distortion function. In recent years, mathematicians have made significant progress in studying inequalities and various properties related to complete elliptic integrals, especially Legendre elliptic integrals and generalized complete elliptic integrals of the first and second types ^{[3,4,5,10,11,23,26]}.

We first introduce some necessary notation. For complex numbers $a, b, c$ with $c \neq 0, -1, -2, \ldots$, and $x \in (-1, 1)$, the Gaussian hypergeometric function ^[9] is defined as follows:

where $(a, n)\equiv a(a+1)\cdots (a+n-1)$ is the shifted factorial function for $n\in\mathbb{N}^+$, and $(a, 0) = 1$ for $a\neq 0$. As we all know, $_2F_1(a, b; c;x)$ has many important applications in the theory of geometric functions and several other contexts ^[19]. Many special functions in mathematical physics are special or limit cases of this function ^[1]. Bhayo studied a new form of the generalized $(p, q)$-complete elliptic integrals as an application of generalized $(p, q)$-trigonometric functions ^[6]. In recent years, the generalization of classical trigonometric functions has attracted significant interest ^[8,20]. For this, we need the generalized arcsine function $\arcsin_{p, q}(x)$ and the generalized $\pi_{p, q}$. For $p, q\in(1, \infty)$, set

and the generalized $\pi_{p, q}$ is the number defined by

where $B$ is the beta function. For ${\rm Re} x > 0$ and ${\rm Re} y > 0$, the classical gamma function $\Gamma(x)$ and beta function $B(x, y)$ are respectively defined as

Clearly, $\arcsin_{p, q}(x)$ is an increasing homeomorphism from $[0, 1]$ onto $[0, \pi_{p, q}/2]$, and its inverse function is the generalized $(p, q)$-sine function $\sin_{p, q}$ is defined on the interval $[0, \pi_{p, q}/2]$. Moreover, the function $\sin_{p, q}$ can be extended to the interval $[0, \pi_{p, q}]$ by

$\sin_{p, q}$ can be also extended to the whole $\mathbb{R}$, and the generalized $(p, q)$-sine function reduces to the classical sine function for $p = q = 2$.

Applying the definitions of $\sin_{p, q}(x)$ and $\pi_{p, q}$, we can define the generalized $(p, q)$-elliptic integrals of the first kind $\mathscr K_{p, q}$ and of the second kind $\mathscr E_{p, q}$ by

and

respectively, for $p, q\in(1, \infty)$, $r\in(0, 1)$.

As a special case of the Gaussian hypergeometric function, these generalized $(p, q)$-elliptic integrals can be represented by Gaussian hypergeometric functions ^[15] as

and

where $p, q\in(1, \infty)$, $r\in(0, 1)$, $r' = \left(1-r^q\right)^{1/q}$. If $p = q = 2$, we can derive the classical complete elliptic integrals $\mathscr K$ and $\mathscr E$, which are well-known complete elliptic integrals of the first kind and second kind, respectively. These complete elliptic integrals play an important role in many branches of quasiconformal mapping, complex analysis, and physics.

In 2017, Yang et al. ^[24] showed that the ratio $\mathscr K(r)/\ln(c/r')$ is strictly concave if and only if $c = e^{4/3}$ on $(0, 1)$, and $\mathscr K(r)/\ln(1+4/r')$ is strictly convex on $(0, 1)$.

In 2019, Wang et al. ^[22] presented the convexity of the function $(\mathscr E'_a-r^2 \mathscr K'_a)/r'^2$ and some properties of the function $\alpha \mathscr K_a(r^{\alpha})$ with respect to the parameter $\alpha$.

In 2020, Huang et al. ^[13] established monotonicity properties for certain functions involving the complete $p$-elliptic integrals of the first and second kinds. They also presented the inequality $\pi/2-\log2+\log(1+1/r')+\alpha(1-r') < \mathscr K_{p}(r) < \pi/2-\log2+\log(1+1/r')+\beta(1-r')$ which holds for all $r\in(0, 1)$ with the best possible constants $\alpha$ and $\beta$. Moreover, these generalized elliptic integrals have significant applications in the theory of geometric functions and in the theory of mean values. More properties and applications of these integrals are given in ^{[7,13,17,21,22,24,25]}.

The generalized $(p, q)$-elliptic integrals of the first kind $\mathscr K_{p, q}$ and of the second kind $\mathscr E_{p, q}$ satisfy the following Legendre relation:

This relation has important applications in many areas of mathematics and physics, including celestial mechanics, quantum mechanics, and statistical mechanics. when $p = q = 2$, the equation reduces to the classical Legendre relation.

Inspired by the papers ^[22], ^[24] and ^[13], we can consider extending the results of $\mathscr K(r)$, $\mathscr K_a(r)$ and $\mathscr K_{p}(r)$ to the generalized $(p, q)$-elliptic integrals of the first kind $\mathscr K_{p, q}$.

A generalized $(p, q)$-modular equation of degree $h > 0$ is

which $a, b, c > 0$ with $a+b\geq c$. Using the decreasing homeomorphism $\mu_{p, q}:(0, 1)\rightarrow (0, \infty)$ defined by

for $p, q\in(1, \infty)$. The function $\mu_{p, q}$ is called the generalized $(p, q)$-Gr$\ddot{o}$tzsch ring function, we can rewrite (1.5) as

The solution of (1.6) is given by

For $p, q\in(1, \infty)$, $r\in(0, 1)$, $K\in(0, \infty)$, we have

The function $\varphi_K^{p, q}(r)$ is referred to as the generalized $(p, q)$-Hersch-Pfluger distortion function with degree $K = 1/h$. For $p = q = 2$, the functions $\mu_{p, q}(r)$ and $\varphi_K^{p, q}(r)$ reduces to well-known special cases that Gr$\ddot{o}$tzsch ring function $\mu(r)$ and Hersch-Pfluger distortion function $\varphi_K(r)$, respectively, which play important role in the theory of plane quasiconformal mappings.

In 2015, Alzer et al. ^[2] studied the monotonicity, convexity, and concavity properties of the function $\mu(r^{\alpha})/\alpha$, and established various Gr$\ddot{o}$tzsch ring functional inequalities based on these properties.

In ^[3], the authors focused on studying the properties of the generalized Gr$\ddot{o}$tzsch ring function $\mu_{a}(r)$ and the generalized Hersch-Pfluger distortion function $\varphi_K^a(r)$. They derived various inequalities involving $\mu_{a}(r)$ and $r^{-1/K}\varphi_K^a(r)$ by utilizing the monotonicity, concavity, and convexity of the functions $\mu_{a}(r)$, $r^{-1/K}\varphi_K^a(r)$, $\log\left(1/\varphi_K^a(e^{-x})\right)$, and $\varphi_K^a(rx)/\varphi_K^a(x)$.

In 2022, Lin et al. ^[16] explored the monotonicity and convexity properties of the function $\mu_{p, q}(r)$ and obtained sharp functional inequalities that sharpen and extend some existing results on the modulus of $\mu(r)$.

Inspired by the papers ^[3], ^[12], and ^[16], our motivation is to extend the existing results of the functions $\varphi_K(r)$ and $\varphi_K^a(r)$ to the generalized $(p, q)$-Hersch-Pfluger distortion function. Our goal is to gain the properties of the generalized $(p, q)$-Hersch-Pfluger distortion function and derive sharp functional inequalities for this function.

Our main objective of this paper is to investigate various properties of the generalized $(p, q)$-elliptic integrals and the generalized $(p, q)$-Hersch-Pfluger distortion function. We specifically focus on establishing complete monotonicity, logarithmic, geometric concavity, and convexity properties of certain functions involving these generalized integrals and arcsine functions. Additionally, they derive several sharp functional inequalities for the generalized $(p, q)$-Hersch-Pfluger distortion function, which improve upon and generalize existing results. Apart from the introduction, this paper consists of three additional sections. Section 2 contains some preliminaries as well as several formulas and lemmas. In Section 3, we present some of the major results regarding the generalized (p, q)-elliptic integrals and provide their proof. In Section 4, we study the generalized (p, q)-Hersch-Pfluger distortion function, and present some of the major results and provide their proof.

2.   Preliminaries

In this section, we present several formulas and lemmas that have been extensively utilized in the paper. These formulas and lemmas play a crucial role in the analysis and proofs of the major results. Throughout this paper, we denote $p, q\in(1, \infty)$, $r\in(0, 1)$ and $r' = \left(1-r^q\right)^{1/q}$.

Lemma 2.1. ^[16] Derivative formulas:

Based on the derivative formula Lemma 2.1, we derive the derivative formulas of the function $\varphi_K^{p, q}(r)$ in the following lemma.

Lemma 2.2. Let $p, q\in(1, \infty)$, then

for $r\in(0, 1)$.

Proof. By the definitions of $s = \varphi_K^{p, q}(r)$, $\mu_{p, q}(s) = \mu_{p, q}(r)/K$, and the derivative formulas of $\mathscr K_{p, q}(r)$, we can derive the following equation

then the derivative formulas of $\varphi_K^{p, q}(r)$ is following. □

Lemma 2.3. ^[4] For $p\in[0, \infty)$, let $I = [0, p)$, and suppose that $f, g:I\rightarrow [0, \infty)$ are functions such that $f(x)/g(x)$ is decreasing on $I\setminus\{0\}$ and $g(0) = 0$, $g(x) > 0$ for $x > 0$. Then

for $x, y, x+y\in I$. Moreover, if the monotoneity of $f(x)/g(x)$ is strict, then the above inequality is also strict on $I\setminus\{0\}$.

The following result is a monotone form of L'H$\hat{o}$pital's Rule ^[4] and will be useful in deriving monotoneity properties and obtaining inequalities.

Lemma 2.4. ^[4] For $-\infty < a < b < \infty$, let $f, g:[a, b]\rightarrow R$ be continuous on $[a, b]$, and be differentiable on $(a, b).$ Let $g'(x)\neq0$ on $(a, b)$. If $f'(x)/g'(x)$ is increasing (decreasing) on $(a, b)$, then so are

If $f'(x)/g'(x)$ is strictly monotone, then the monotonicity in the conclusion is also strict.

The following lemma presents some known results of generalized $(p, q)$-elliptic integrals, which can be utilized to prove the main results of this paper.

Lemma 2.5. ^[14] For $p, q\in(1, \infty)$, $r\in(0, 1)$, $a = 1-1/p, b = a+1/q$, then the functions

(1) $h_1(r) = \dfrac{ \mathscr E_{p, q}(r)-r'^q \mathscr K_{p, q}(r)}{r^q}$ is strictly increasing and convex from $(0, 1)$ onto $(a\pi_{p, q}/(2b), 1)$.

(2) $h_2(r) = \dfrac{ \mathscr E_{p, q}(r)-r'^q \mathscr K_{p, q}(r)}{r^q \mathscr K_{p, q}(r)}$ is strictly decreasing from $(0, 1)$ onto $(0, a/b)$.

(3) $h_3(r) = r'^c \mathscr K_{p, q}(r)$ is decreasing(increasing) on $(0, 1)$ iff $c\geq a/b$ ($c\leq0$ respectively) with $h_3((0, 1)) = (0, \pi_{p, q}/2)$ if $c\geq a/b$.

(4) $h_4(r) = \dfrac{ \mathscr K_{p, q}(r)- \mathscr E_{p, q}(r)}{r^q \mathscr K_{p, q}(r)}$ is increasing from $(0, 1)$ onto $(1/(qb), 1)$.

(5) $h_5(r) = r'^q \mathscr K_{p, q}(r)/ \mathscr E_{p, q}(r)$ is strictly decreasing from $(0, 1)$ onto itself.

Lemma 2.6. For $r\in (0, 1)$, $K, p, q\in(1, \infty)$, let $s = \varphi_K^{p, q}(r), t = \varphi_{1/K}^{p, q}(r)$.

(1) The function $f(r) = \mathscr K_{p, q}(s)/ \mathscr K_{p, q}(r)$ is increasing from $(0, 1)$ onto $(1, K)$.

(2) For $q > \frac{3p-4}{p-1}$, the function $g(r) = \dfrac{s'^{q/2} \mathscr K_{p, q}(s)^2}{r'^{q/2} \mathscr K_{p, q}(r)^2}$ is decreasing from $(0, 1)$ onto $(0, 1)$.

Proof. (1) According to the Lemma 2.1, we have

Denote

By applying Lemma 2.5(1)(3) and considering $s > r$, we can conclude that $f_1(r)$ is increasing. Hence, we can determine that $f'(r)$ is positive. Therefore, we can deduce that $f(r)$ is an increasing function. For the limiting values, we have $\lim\limits_{r\rightarrow0^+}f(r) = 1$ and $\lim\limits_{r\rightarrow1^-}f(r) = K$.

(2) Let

By differentiation, we have

If $q > \frac{3p-4}{p-1}$, $g_1(r)$ is increasing by Lemma 2.5(2) and (3), thus $g'(r)$ is negative for $s > r$. Hence $g(r)$ is decreasing, the limiting values follow from the definitions (1.2) and (1.7)

□

Lemma 2.7. For $r\in(0, 1)$ and $p, q\in(1, \infty)$, the inequality

holds.

Proof. Let $M(r) = r^qK_{p, q}(r)^2/N(r)$, $M_1(r) = N(r)/(r^qK_{p, q}(r))$, we have

According to Lemma 2.5, we have

and

□

Lemma 2.8. For $r\in(0, 1)$, $p, q\in(1, \infty)$, the function

is strictly increasing from $(0, 1)$ onto $(0, \infty)$.

Proof. By differentiation, we have

where $N(r) = \dfrac{q}{p} \mathscr K_{p, q}(r)\left(\mathscr E_{p, q}(r)-r'^q \mathscr K_{p, q}(r)\right)$+$\mathscr E_{p, q}(r)\left(\mathscr K_{p, q}(r)- \mathscr E_{p, q}(r)\right)-\left(\dfrac{q}{p}-q+1\right)r^q \mathscr K_{p, q}(r) \mathscr E_{p, q}(r)$.

Let $h_2(r) = r^{q/2}\log(1/r)$, $h_3(r) = r'^q$, then $h_1(r) = h_2(r)/h_3(r)$, and $h_2(1^-) = h_3(1^-) = 0$.

By Lemma 2.4, the function $h_1(r)$ is strictly increasing from $(0, 1)$ onto $(0, 1/q)$. According to Lemma 2.7, we conclude that $r^qK_{p, q}(r)^2/N(r) > 1/q$. Therefore, it is easy to check that $h(r)$ is increasing. For the limiting values, $\lim\limits_{r\rightarrow0^+}h(r) = 0$. Since

we have

□

3.   Generalized $(p, q)$-elliptic integrals

In this section, we present some of the main results regarding the generalized $(p, q)$-elliptic integrals.

Theorem 3.1. For $p, q\in(1, \infty)$, the function $F_{p, q}(r) = (\mathscr E'_{p, q}(r)-r^q \mathscr K'_{p, q}(r))/r'^q$ is concave on $(0, r_0^*)$ and convex on $(r_0^*, 1)$ for some point $r_0^*\in(0, 1)$.

Proof. Let $F(r) = (\mathscr E_{p, q}(r)-r'^q \mathscr K_{p, q}(r))/r^q$, by the definitions (1.2) and (1.3), which can be expressed as

with $a = 1-1/p, b = 1/q$. This implies that

By differentiation, we have

Hence,

According to [4, Theorem 1.19(10)],

we have

Let

it is easy to obtain that $F_1(x)$ is strictly increasing from $(0, 1)$ onto $(1, \infty)$, since

Similarly, we can deduce that

is strictly decreasing from $(0, 1)$ onto $\left(\frac{1-b(2a+b+2)}{(1-b)(a+b+2)}, \infty\right)$. Therefore, the sign of $F''_{p, q}(r)$ changes from negative to positive on $(0, 1)$ by (3.1), we know that there exists $r_0^*\in(0, 1)$ such that $F_{p, q}(r)$ is concave on $(0, r_0^*)$ and convex on $(r_0^*, 1)$. □

Theorem 3.2. For $p, q\in(1, \infty)$, $r\in(0, 1)$, $\alpha > 0$, Let $H_{p, q}(\alpha) = \alpha \mathscr K_{p, q}(r^{\alpha})$, $G_{p, q}(\alpha) = \mathscr K_{p, q}(r^{\alpha})/\alpha$. Then

(1) The function $\alpha\mapsto H_{p, q}(\alpha)$ is strictly increasing and log-concave on $(0, \infty)$;

(2) The function $\alpha\mapsto1/H_{p, q}(\alpha)$ is strictly convex on $(0, \infty)$;

(3) The function $\alpha\mapsto G_{p, q}(\alpha)$ is strictly decreasing and log-convex on $(0, \infty)$.

Proof. (1) Let $t = r^{\alpha}$, then ${\rm d}t/{\rm d}\alpha = t\log r < 0$, and

Hence, the monotonicity of $H_{p, q}(\alpha)$ follows from Lemma 2.8.

By logarithmic differentiation,

It is not difficult to verify that ${\rm d}(\log H_{p, q}(\alpha))/{\rm d}\alpha$ is strictly increasing with respect to $t$ by Lemma 2.8, and is strictly decreasing with respect to $\alpha$. Thus the function $\alpha\mapsto H_{p, q}(\alpha)$ is strictly increasing and log-concave on $(0, \infty)$.

(2) Since $t = r^{\alpha}$, then $\alpha = \log(1/t)/\log(1/r)$. Differentiating $1/H_{p, q}(\alpha)$ yields

where $h(t)$ is defined in Lemma 2.8. Let $f(r) = \log(1/r)K_{p, q}(r)$, $f_1(r) = \log(1/r)$, $f_2(r) = 1/K_{p, q}(r)$, We clearly see that $f_1(1) = f_2(1) = 0$, then

which is decreasing follows from Lemma 2.5(1), (3). Hence the function $f(r)$ is decreasing from $(0, 1)$ onto $(0, \infty)$ by Lemma 2.4. Therefore, it follows from Lemma 2.8 and (3.2) that the function $\alpha\mapsto1/H_{p, q}(\alpha)$ is strictly convex on $(0, \infty)$.

(3) Since $t = r^{\alpha}$, and ${\rm d}t/{\rm d}\alpha = t\log r < 0$, simple computations yields

Let $g(r) = \dfrac{ \mathscr E_{p, q}(r)-r'^q \mathscr K_{p, q}(r)}{r'^qK_{p, q}(r)}+\dfrac{1}{\log(1/r)}$, we have

is positive by Lemma 2.8, where the defition of $N(r)$ is in Lemma 2.8. Hence the function $g(r)$ is increasing from $(0, 1)$ onto $(0, \infty)$. Since $\log r < 0$, the monotonicity of the function $G_{p, q}(\alpha)$ follows immediately.

Since $\log G_{p, q}(\alpha) = \log \mathscr K_{p, q}(t)-\log\alpha$, by differentiation, we obtain that

It follows from (3.3) and Lemma 2.5(5) that ${{\rm d}\left(\log G_{p, q}(\alpha)\right)}/{{\rm d}\alpha}$ is strictly decreasing with respect to $t$. Therefore, $G_{p, q}(\alpha)$ is log-convex on $(0, \infty)$ with respect to $\alpha$. □

Next, we apply Theorem 3.2 to obtain the inequality involving the generalized $(p, q)$-elliptic integrals $\mathscr K_{p, q}$.

Corollary 3.1. For $p, q\in(1, \infty)$.

(1) Let $\alpha, \beta$ be positive numbers with $\alpha > \beta > 0$. The double inequality

holds for all $r\in(0, 1)$.

(2) Inequality

holds with equality if and only if $x = y$ for all $x, y\in(0, 1)$.

(3) Inequality

holds with equality if and only if $x = y$ for all $x, y\in(0, 1)$.

(4) Let $\alpha, \beta$ be positive numbers with $\alpha > \beta > 0$. The double inequality

holds for all $r\in(0, 1)$.

(5) Inequality

holds with equality if and only if $x = y$ for all $x, y\in(0, 1)$.

Proof. (1) By utilizing the monotonicity of the function $H_{p, q}(\alpha)$ stated in Theorem 3.2, along with the monotonicity of the function $\mathscr K_{p, q}(r)$, we can establish that $\alpha \mathscr K_{p, q}(r^{\alpha}) > \beta \mathscr K_{p, q}(r^{\beta})$. Consequently, the double inequality holds.

(2) Since the function $\alpha\mapsto H_{p, q}(\alpha)$ is strictly log-concave on $(0, \infty)$, we can deduce that

with equality if and only if $\alpha = \beta$ for $\alpha, \beta > 0$. For $x, y\in(0, 1)$ and set

Simple computations yields

Hence, the inequality

hold with equality if and only if $x = y$.

(3) Since the function $\alpha\mapsto1/H_{p, q}(\alpha)$ is strictly convex on $(0, \infty)$, we get

with equality if and only if $\alpha = \beta$ for $\alpha, \beta > 0$. Set

From (3.4)–(3.6), we conclude that the inequality hold with equality if and only if $x = y$.

(4) Since the function $\alpha\mapsto G_{p, q}(\alpha)$ is strictly decreasing, and the monotonicity of the function $\mathscr K_{p, q}(r)$, we have

(5) Since the function $\alpha\mapsto G_{p, q}(\alpha)$ is log-convex on $(0, \infty)$,

with equality if and only if $\alpha = \beta$ for $\alpha, \beta > 0$. Set

Simple computations yields

Hence, the inequality

hold with equality if and only if $x = y$. □

Remark 3.1. In ^[22], Wang et al. provided the proof for the convexity of the function $(\mathscr E'_a-r^2 \mathscr K'_a)/r'^2$ and presented certain properties of the functions $\alpha \mathscr K_a(r^{\alpha})$ and $1/\alpha \mathscr K_a(r^{\alpha})$ with respect to the parameter $\alpha$. It is worth noting that Theorem 3.1 and Theorem 3.2(1), (2) can be reduced to [22, Theorem 1.1, Theorem 1.3] if $p = q = 1/a$.

4.   Generalized $(p, q)$-Hersch-Pfluger distortion function

In this section, we study the complete monotonicity, logarithmic, geometric concavity and convexity of the generalized $(p, q)$-Hersch-Pfluger distortion function, and present some of the main results about $\varphi_K^{p, q}$.

Theorem 4.1. For $K, p, q\in(1, \infty)$, and $q > \frac{3p-4}{p-1}$, let $a = 1-1/p, b = 1/q$, $f, g$ be defined on $(0, 1]$ by

Then $f$ is decreaseing from $(0, 1]$ onto $\left[1, e^{bR(a, b)\left(1-1/K\right)}\right)$, and $g$ is increasing from $(0, 1]$ onto $\left(e^{bR(a, b)(1-K)}, 1\right]$.

Proof. Let $s = \varphi_K^{p, q}(r)$, then $f(r) = \dfrac{s}{r^{1/K}}$, we have

Hence,

According to Lemma 2.6(2), $f'(r)$ is negative. Combine with $f(1^-) = 1$ and by [18, Theorem 2],

Let $t = \varphi_{1/K}^{p, q}(r)$, thus $r = \varphi_{K}^{p, q}(t)$ and

According to the monotonicity of $f(r)$, $g(r)$ is increasing on $(0, 1]$. The limiting values can also be derived from [18, Theorem 2]. □

Next, we utilize Theorem 4.1 to derive the inequality concerning the generalized $(p, q)$-Hersch-Pfluger distortion function $\varphi_K^{p, q}$.

Corollary 4.1. For $K, p, q\in(1, \infty)$, and $q > \frac{3p-4}{p-1}$, let $a = 1-1/p, b = 1/q$, then

(1) The double inequality

hold with equality if and only if $r = s$.

(2) The double inequality

hold with equality if and only if $r = s$.

Proof. (1) According to Lemma 2.3 and the monotonicity of the function $f(r) = r^{-1/K}\varphi_K^{p, q}(r)$, we can conclude that

for $x, y\in(0, 1)$. Set $r = x+y$ and $s = y$, we get

According to $f(r)$ is decreaseing from $(0, 1]$ onto $\left[1, e^{bR(a, b)\left(1-1/K\right)}\right)$, we obtain that

with equality if and only if $r = s$.

(2) By Lemma 2.3 and the monotonicity of $g(r)$, we have

for $x, y\in(0, 1)$. Set $r = x+y$ and $s = y$, we obtain that

with equality if and only if $r = s$.

According to $g(r)$ increasing from $(0, 1]$ onto $\left(e^{bR(a, b)(1-K)}, 1\right]$, we get

with equality if and only if $r = s$. Therefore, the double inequality (4.2) hold. □

Theorem 4.2. For $K, p, q\in(1, \infty)$, $q > \frac{3p-4}{p-1}$, the function $f(x) = \log\left(1/\varphi_K^{p, q}(e^{-x})\right)$ is increasing and convex on $(0, \infty)$, $g(x) = \log\left(1/\varphi_{1/K}^{p, q}(e^{-x})\right)$ is increasing and concave on $(0, \infty)$, and

with equality if and only if $K = 1$ for each $r, t\in(0, 1)$.

Proof. Let $r = e^{-x}, s = \varphi_K^{p, q}(r)$, according to the Lemma 2.6, we have

is positive and increasing with respect to $x$. Thus $f$ is increasing and convex. Therefore,

and putting $r = e^{-x}, t = e^{-y}$, we obtain

with equality if and only if $K = 1$ for each $r, t\in(0, 1)$. The proof for $g(x)$ follows a similar approach. □

Theorem 4.3. For $K, p, q\in(1, \infty)$, $q > \frac{3p-4}{p-1}$, $r\in(0, 1)$, the function $f(x) = \varphi_K^{p, q}(rx)/\varphi_K^{p, q}(x)$ is increasing from $(0, 1)$ onto $\left(r^{1/K}, \varphi_K^{p, q}(r)\right)$, while function $g(x) = \varphi_{1/K}^{p, q}(rx)/\varphi_{1/K}^{p, q}(x)$ is decreasing from $(0, 1)$ onto $\left(\varphi_{1/K}^{p, q}(r), r^K\right)$. In particular,

with equality if and only if $K = 1$ for each $r, t\in(0, 1)$.

Proof. Let $t = rx, u = \varphi_K^{p, q}(t), s = \varphi_K^{p, q}(x)$,

then

Since $t < x$ and $\dfrac{s'^{q/2} \mathscr K_{p, q}(s)}{x'^{q/2} \mathscr K_{p, q}(x)}$ is decreasing with respect to $r$ by Lemma 2.6, $f'(x)$ is positive on $(0, 1)$. For the limiting values, by using L'H$\hat{o}$pital's Rule, we get

Since the monotonicity of the function $f(x)$, along with the definition of the function $\varphi_K^{p, q}(r)$, we obtain

with equality if and only if $K = 1$ for each $r, t\in(0, 1)$. The proof for $g(x)$ follows a similar approach. As a result, we will omit the detailed proof. □

Theorem 4.4. For $K, p\in(1, \infty)$, $r\in(0, 1)$, $q > 3$ and $p > \frac{q(q-3)}{q^2-2q-1}$, the function $f(r)$ defined by

is strictly decreasing from $(0, 1]$ into $\left[1, e^{(1-1/K)bR(a, b)}\right)$, the function $g(r)$ defined by

is strictly increasing from $(0, 1]$ into $\left(e^{(1-K)bR(a, b)}, 1\right]$, where $a = 1-1/p, b = a+1/q$.

Proof. Let $s = \varphi_K^{p, q}(r)$, $f_1(r) = \arcsin(s)$, $f_2(r) = \arcsin\left(r^{1/K}\right)$, $f(r) = f_1(r)/f_2(r)$. Since $f_1(0) = f_2(0) = 0$, we have

Let $f_3(r) = \dfrac{s'^{q-1} \mathscr K_{p, q}(s)^2}{r'^{q-1} \mathscr K_{p, q}(r)^2}$, according to the Lemma 2.1, we have

where

Using Lemma 2.5(2) and (3), we can observe that $f_4(r)$ is positive. Consequently, we can deduce that $f_3(r)$ is decreasing. As a result, we obtain that $f(r)$ is decreasing on the interval $(0, 1]$ by Theorem 4.1. Furthermore, it can be deduced that the monotonicity of $g(r)$ is similar to that of $f(r)$. □

Remark 4.1. Theorems 4.1 to 4.4 can be seen as variations and extensions of the results presented in [3, Theorem 1.14, Theorem 1.15, Theorem 6.7, Theorem 6.13]. When $p = q = 1/a$, the results obtained in Theorems 4.1 to 4.4 can be reduced to those obtained in ^[3].

5.   Conclusions

In this paper, we investigate the properties of the generalized $(p, q)$-elliptic integrals and the generalized $(p, q)$-Hersch-Pfluger distortion function. Through our analysis, we have established complete monotonicity, logarithmic, geometric concavity, and convexity properties for certain functions involving these integrals and arcsine functions. These properties provide valuable insights into the behavior of these functions. Furthermore, we have derived several sharp functional inequalities for the generalized $(p, q)$-elliptic integrals and the generalized $(p, q)$-Hersch-Pfluger distortion function. These inequalities not only improve upon existing results but also generalize them.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence tools in the creation of this article.

Acknowledgments

The research was supported by the Natural Science Foundation of China (Grant Nos. 11601485, 11401531).

Conflict of interest

The authors declare that they have no conflicts of interest.