Edge-coloring of generalized lexicographic product of graphs

1.   Introduction

In 1998, Kahlig and Matkowski ^[1] proved in particular that every homogeneous bivariable mean $\mathbf{M}$ in $(0, \infty)$ can be represented in the form

where $\mathbf{A}$ is the arithmetic mean and $f_{\mathbf{M, A}}$: $\left(-1, 1\right) \longrightarrow \left(0, 2\right)$ is a unique single variable function (with the graph laying in a set of a butterfly shape), called an $\mathbf{A}$-index of $\mathbf{M}$.

In this paper we consider Seiffert function $f:(0, 1)\rightarrow \mathbb{R}$ which fulfils the following condition

According to the results of Witkowski ^[2] we introduce the mean $\mathbf{M}_{f}$ of the form

In this paper a mean $\mathbf{M}_{f}:\mathbb{R}_{+}^{2}\longrightarrow \mathbb{R}$ is the function that is symmetric, positively homogeneous and internal in sense ^[2]. Basic result of Witkowski is correspondence between a mean $\mathbf{M}_{f}$ and Seiffert function $f$ of $\mathbf{M}_{f}$ is given by the following formula

where

Therefore, $f$ and $\mathbf{M}_{f}$ form a one-to-one correspondence via $\left(1.1\right)$ and $\left(1.2\right)$. For this reason, in the following we can rewrite $f = :f_{\mathbf{M}}$.

Throughout this article, we say $x\neq y$, that is, $t\in (0, 1)$. For convenience, we note that $\mathbf{M}_{1} < \mathbf{M}_{2}$ means $\mathbf{M} _{1}(x, y) < \mathbf{M}_{2}(x, y)$ holds for two means $\mathbf{M}_{1}$ and $\mathbf{M}_{2}$ with $x\neq y$. Then there is a fact that the inequality $f_{ \mathbf{M}_{1}}(t) > f_{\mathbf{M}_{2}}(t)$ holds if and only if $\mathbf{M} _{1} < \mathbf{M}_{2}$. That is to say,

The above relationship $\left(1.4\right)$ inspires us to ask a question: Can we transform the means inequality problem into the reciprocal inequality problem of the corresponding Seiffert functions? Witkowski ^[2] answers this question from the perspective of one-to-one correspondence. We find that these two kinds of inequalities are equivalent in similar linear inequalities. We describe this result in Lemma 2.1 as a support of this paper.

As we know, the study of inequalities for mean values has always been a hot topic in the field of inequalities. For example, two common means can be used to define some new means. The recent success in this respect can be seen in references ^{[3,4,5,6,7,8]}. In ^[2], Witkowski introduced the following two new means, one called sine mean

and the other called hyperbolic tangent mean

Recently, Nowicka and Witkowski ^[9] determined various optimal bounds for the $\mathbf{M}_{\sin }(x, y)$ and $\mathbf{M}_{\tanh }(x, y)$ by the arithmetic mean $\mathbf{A}(x, y) = (x+y)/2$ and centroidal mean

as follows:

Proposition 1.1. The double inequality

holds if and only if $\alpha \leq$ $1/2$ and $\beta \geq \left(3/\sin 1\right) -3\thickapprox 0.5652.$

Proposition 1.2. The double inequality

holds if and only if $\alpha \leq$ $\left(3/\tanh 1\right) -3 \thickapprox 0.9391$ and $\beta \geq 1.$

Proposition 1.3. The double inequality

holds if and only if $\alpha \leq$ $4\sin 1-3\thickapprox 0.3659$ and $\beta \geq 1/2.$

Proposition 1.4. The double inequality

holds if and only if $\alpha \leq$ $0$ and $\beta \geq 4\tanh 1-3\thickapprox 0.0464.$

Proposition 1.5. The double inequality

holds if and only if $\alpha \leq$ $1/2$ and $\beta \geq \left(9\cot ^{2}1\right) /7\thickapprox 0.5301.$

Proposition 1.6. The double inequality

holds if and only if $\alpha \leq$ $\left(9\left(\coth ^{2}1-1\right) \right) /7 \thickapprox 0.9309$ and $\beta \geq 1.$

Proposition 1.7. The double inequality

holds if and only if $\alpha \leq \left(16\sin ^{2}1-9\right) /7\thickapprox 0.3327$ and $\beta \geq 1/2.$

Proposition 1.8. The double inequality

holds if and only if $\alpha \leq 0$ and $\beta \geq \left(16\tanh ^{2}1-9\right) /7\thickapprox 0.0401.$

In essence, the above results are how the two new means $\mathbf{M}_{\sin }$ and $\mathbf{M}_{\tanh }$ are expressed linearly, harmoniously, squarely, and harmoniously in square by the two classical means $\mathbf{C}_{e}(x, y)$ and $\mathbf{A}(x, y)$. In this paper, we study the following two-sided inequalities in exponential form for nonzero number $p\in \mathbb{R}$

in order to reach a broader conclusion including all the above properties. The main conclusions of this paper are as follows:

Theorem 1.1. Let $x, y > 0$, $x\neq y$, $p\neq 0$ and

Then the following are considered.

(i) If $p\geq p^{\clubsuit }$, the double inequality

holds if and only if $\alpha _{p}\leq 3^{p}\left(1-\sin ^{p}1\right) /\left[\left(\sin ^{p}1\right) \left(4^{p}-3^{p}\right) \right]$ and $\beta _{p}\geq 1/2$.

(ii) If $0 < p\leq 12/5$, the double inequality

holds if and only if $\alpha _{p}\leq 1/2$ and $\beta _{p}\geq 3^{p}\left(1-\sin ^{p}1\right) /\left[\left(\sin ^{p}1\right) \left(4^{p}-3^{p}\right) \right]$.

(iii) If $p < 0$, the double inequality

holds if and only if $\alpha _{p}\leq 1/2$ and $\beta _{p}\geq 3^{p}\left(1-\sin ^{p}1\right) /\left[\left(\sin ^{p}1\right) \left(4^{p}-3^{p}\right) \right]$.

Theorem 1.2. Let $x, y > 0$, $x\neq y$, $p\neq 0$ and

Then the following are considered:

(i) If $p > 0$, the double inequality

holds if and only if $\lambda _{p}\leq \left(\left(\coth 1\right) ^{p}-1\right) /\left(\left(4/3\right) ^{p}-1\right)$ and $\mu _{p}\geq 1.$

(ii) If $p^{\ast }\leq p < 0$,

holds if and only if $\lambda _{p}\leq \left(\left(\coth 1\right) ^{p}-1\right) /\left(\left(4/3\right) ^{p}-1\right)$ and $\mu _{p}\geq 1.$

2.   Lemmas

We first introduce a theoretical support of this paper.

Lemma 2.1. (^[10]) Let $\mathbf{K}(x, y), \mathbf{R}(x, y),$ and $\mathbf{N}(x, y)$ be three means with two positive distinct parameters $x$ and $y;$ $f_{\mathbf{K}}(t),$ $f_{ \mathbf{R}}(t),$ and $f_{\mathbf{N}}(t)$ be the corresponding Seiffert functions of the former, $\vartheta _{1}, \vartheta _{2}, \theta _{1}, \theta _{2}, p\in \mathbb{R},$ and $p\neq 0$. Then

It must be mentioned that the key steps to prove the above results are following:

where

and $0 < t < 1,$

In order to prove the main conclusions, we shall introduce some very suitable methods which are called the monotone form of L'Hospital's rule (see Lemma $2.2$) and the criterion for the monotonicity of the quotient of power series (see Lemma $2.3$).

Lemma 2.2. (^[11,12]) For $-\infty < a < b < \infty$, let $f, g:[a, b]\rightarrow \mathbb{R}$ be continuous functions that are differentiable on $\left(a, b\right)$, with $f\left(a\right) = g\left(a\right) = 0$ or $f\left(b\right) = g\left(b\right) = 0$. Assume that $g^{\prime }(t)\neq 0$ for each $x$ in $(a, b)$. If $f^{\prime }/g^{\prime }$ is increasing (decreasing) on $(a, b)$, then so is $f/g$.

Lemma 2.3. (^[13]) Let $a_{n}$ and $b_{n}$ $(n = 0, 1, 2, \cdot \cdot \cdot)$ be real numbers, and let the power series $A(x) = \sum_{n = 0}^{\infty }a_{n}x^{n}$ and $B(x) = \sum_{n = 0}^{\infty }b_{n}x^{n}$ be convergent for $\left\vert x\right\vert < R$ $(R\leq +\infty)$. If $b_{n} > 0$ for $n = 0, 1, 2, \cdot \cdot \cdot$, and if $\varepsilon _{n} = a_{n}/b_{n}$ is strictly increasing (or decreasing) for $n = 0, 1, 2, \cdot \cdot \cdot$, then the function $A(x)/B(x)$ is strictly increasing (or decreasing) on $(0, R)$ $(R\leq +\infty)$.

Lemma 2.4. (^[14,15]) Let $B_{2n}$ be the even-indexed Bernoulli numbers. Then we have the following power series expansions

Lemma 2.5. (^{[16,17,18,19,20]}) Let $B_{2n}$ the even-indexed Bernoulli numbers, $n = 1, 2, \ldots$. Then

Lemma 2.6. Let $l_1(t)$ be defined by

where

Then the double inequality

holds for all $t\in (0, 1)$, where the constants $12/5$ and $\left(3\cos 2+\sin 2+1\right) /\left(3\sin 2-\cos 2-3\right) \thickapprox 4.\, 588$ are the best possible in (2.6).

Proof. Since

and

where

where

Setting

we have

Here we prove that the sequence $\{q_{n}\}_{n\geq 2}$ decreases monotonously. Obviously, $q_{2} > 0 > q_{3}$. We shall prove that for $n\geq 3,$

that is,

By Lemma 2.5 we have

and

So we can complete the prove $\left(2.7\right)$ when proving

or

In fact,

where

for all $n\geq 3$.

So the sequence $\{q_{n}\}_{n\geq 2}$ decreases monotonously. By Lemma 2.3 we obtain that $r_{1}(t)/s_{1}(t)$ is decreasing on $(0, 1)$, which means that the function $l_{1}\left(t\right)$ is increasing on $(0, 1)$. In view of

the proof of this lemma is complete.

Lemma 2.7. Let $l_{2}(t)$ be defined by

where

Then $l_{2}(t)$ is strictly decreasing on $\left(0, \infty \right)$.

Proof. Let's take the power series expansions

into $A(t)$ and $B(t)$, and get

where

Setting

Here we prove that the sequence $\{k_{n}\}_{n\geq 2}$ decreases monotonously. Obviously, $k_{2} = 0 < k_{3}$. For $n\geq 3,$

follows from $\Delta (n) > 0$ for all $n\geq 2$, where

with

We have that $\Delta (2) = 5376 > 0$ and shall prove that

holds for all $n\geq 3$. Now we use mathematical induction to prove $(2.8)$. When $n = 3$, the left-hand side and right-hand side of $(2.8)$ are $2^{6} = 64$ and $i(3)/j(3) = 941/17\thickapprox 55.\, 353$, which implies $(2.8)$ holds for $n = 3$. Assuming that $(2.8)$ holds for $n = m$, that is,

Next, we prove that $(2.8)$ is valid for $n = m+1$. By $(2.9)$ we have

in order to complete the proof of $(2.8)$ it suffices to show that

In fact,

for $m\geq 3$ due to the coefficients of the power square of $\left(m-1\right)$ are positive.

By Lemma 2.3 we get that $A(t)/B(t)$ is strictly increasing on $\left(0, \infty \right).$ So the function $l_{2}(x)$ is strictly decreasing on $\left(0, \infty \right)$.

The proof of Lemma $2.7$ is complete.

3.   Proofs of main results

Via (1.3) and (1.2) we can obtain

Then by Lemma 2.1 and $\left(2.3\right)$ we have

So we turn to the proof of the following two theorems.

Theorem 3.1. Let $t\in (0, 1)$ and

Then,

(i) if $p\geq p^{\clubsuit }$, the double inequality

holds if and only if $\alpha _{p}\leq 3^{p}\left(1-\sin ^{p}1\right) /\left[\left(\sin ^{p}1\right) \left(4^{p}-3^{p}\right) \right]$ and $\beta _{p}\geq 1/2$;

(ii) if $0\neq p\leq 12/5 = 2.\, 4$ and $p\neq 0$ the double inequality

holds if and only if $\alpha _{p}\leq 1/2$ and $\beta \geq 3^{p}\left(1-\sin ^{p}1\right) /\left[\left(\sin ^{p}1\right) \left(4^{p}-3^{p}\right) \right].$

Theorem 3.2. Let $t\in (0, 1)$ and

If $0\neq p\geq - 3.\, 477\, 6$, the double inequality

holds if and only if $\lambda _{p}\leq \left(\left(\coth 1\right) ^{p}-1\right) /\left(\left(4/3\right) ^{p}-1\right)$ and $\mu _{p}\geq 1$.

3.1.   The proof of Theorem 3.1

Let

where

Then

and

where the three functions $s_{1}(t)$, $r_{1}(t)$, and $l_{1}(t)$ are shown in Lemma $2.6$.

By Lemma $2.6$ we can obtain the following results:

(a) When $p\geq \max_{t\in (0, 1)}l_{1}(t) = :p^{\clubsuit } = \left(3\cos 2+\sin 2+1\right) /\left(3\sin 2-\cos 2-3\right) \thickapprox 4.\, 588$,

this leads to $F(t) = f(t)/g(t)$ is decreasing on $(0, 1)$ by Lemma $2.1$. In view of

we have that $\left(3.1\right)$ holds.

(b) When $0\neq p\leq 12/5 = \min_{t\in (0, 1)}l_{1}(t),$

this leads to $F(t) = f(t)/g(t)$ is increasing on $(0, 1)$ by Lemma $2.2$. In view of $\left(3.4\right)$ we have that $\left(3.2\right)$ holds.

The proof of Theorem 3.1 is complete.

3.2.   The proof of Theorem 3.2

Let

Then

and

where the three functions $A(t)$, $B(t)$, and $l_{2}(t)$ are shown in Lemma $2.7$. By Lemma $2.7$ we see that $l_{2}(x)$ is strictly decreasing on $\left(0, 1\right)$. Since

we obtain the following result:

When $p\geq \max_{t\in \left(0, 1\right) }\left\{ -l_{2}(t)\right\} = -p_{\#} = :p^{\ast }\thickapprox -3.\, 477\, 6,$

this leads to $G(t) = u(t)/v(t)$ is decreasing on $(0, 1)$ by Lemma $2.2$. Since

we have

which completes the proof of Theorem 3.2.

4.   Corollaries of main results and remarks

Remark 4.1. Letting $p = 1, -1, 2, -2$ in Theorems 1.1 and 1.2 respectively, one can obtain Propositions 1.1–1.8.

From Theorems 1.1 and 1.2, we can also get the following important conclusions:

Corollary 4.1. Let $x, y > 0$, $x\neq y$, and

Then the double inequality

holds, where the constants $\alpha$ and $\beta$ are the best possible in (4.1).

Corollary 4.2. Let $x, y > 0$, $x\neq y$, and

Then the double inequality

holds, where the constants $\theta$ and $\vartheta$ are the best possible in (4.2).

Corollary 4.3. Let $x, y > 0$, $x\neq y$, and

Then the double inequality

holds, where the constants $\lambda$ and $\mu$ are the best possible in (4.3).

5.   Conclusions

In this paper, we have studied exponential type inequalities for $\mathbf{M} _{\sin }$ and $\mathbf{M}_{\tanh }$ in term of $\mathbf{A}$ and $\mathbf{Ce}$ for nonzero number $p\in \mathbb{R}$:

obtained a lot of interesting conclusions which include the ones of the previous similar literature. In fact, we can consider similar inequalities for dual means of the two means $\mathbf{M}_{\sin }$ and $\mathbf{M}_{\tanh }$, and we can replace $\mathbf{A}$ and $\mathbf{Ce}$ by other famous means. Therefore, the content of this research is very extensive.

Acknowledgements

The authors are grateful to editor and anonymous referees for their careful corrections to and valuable comments on the original version of this paper.

The first author was supported by the National Natural Science Foundation of China (no. 61772025). The second author was supported in part by the Serbian Ministry of Education, Science and Technological Development, under projects ON 174032 and III 44006.

Conflict of interest

The authors declare that they have no conflict of interest.