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Sharp inequalities between means
Abstract
For p∈ℝ, the p-th power mean M p (a,b), arithmetic mean A(a,b), geometric mean G(a,b), and harmonic mean H(a,b) of two positive numbers a and b are defined by M p (a,b)=a p +b p w,p≠0,ab,p=0, A(a,b)=(a+b)/2, G(a,b)=ab, and H(a,b)=2ab/(a+b), respectively. In this paper, we answer the questions: For α∈(0,1), what are the greatest values p, r and m, and the least values q, s and n, such that the inequalities M p (a,b)≤A α (a,b)G 1-α (a,b)≤M q (a,b), M r (a,b)≤G α (a,b)H 1-α (a,b)≤M s (a,b) and M m (a,b)≤A α (a,b)H 1-α (a,b)≤M n (a,b) hold for all a,b>0?
... In recent years, it has become clear that math well-known inequalities using a variety of cutting-e erators. The books [55][56][57][58][59][60][61][62][63][64][65][66][67] mentioned in this contex tiated the exploration of the concept of a harmonic s of inequalities. He introduced the classical Hermite cally convex functions. ...
... In recent years, it has become clear that mathematicians strongly prefer to present well-known inequalities using a variety of cutting-edge theories of fractional integral operators. The books [55][56][57][58][59][60][61][62][63][64][65][66][67] mentioned in this context may be consulted. Işcan [68] has ini-tiated the exploration of the concept of a harmonic set and finds its application in the field of inequalities. ...
... In recent years, it has become clear that mathematicians strongly prefer to present well-known inequalities using a variety of cutting-edge theories of fractional integral operators. The books [55][56][57][58][59][60][61][62][63][64][65][66][67] mentioned in this context may be consulted. Işcan [68] has initiated the exploration of the concept of a harmonic set and finds its application in the field of inequalities. ...
In this paper, we provide different variants of the Hermite–Hadamard (H⋅H) inequality using the concept of a new class of convex mappings, which is referred to as up and down harmonically s-convex fuzzy-number-valued functions (UDH s-convex FNVM) in the second sense based on the up and down fuzzy inclusion relation. The findings are confirmed with certain numerical calculations that take a few appropriate examples into account. The results deal with various integrals of the 2ρσρ+σ type and are innovative in the setting of up and down harmonically s-convex fuzzy-number-valued functions. Moreover, we acquire classical and new exceptional cases that can be seen as applications of our main outcomes. In our opinion, this will make a significant contribution to encouraging more research.
... The following theorem provides a criteria for a function to be 4-convex. Due to the massive properties and consequences of convex functions, a lot of problems have been solved and modeled in diverse fields of science with the help of this class of functions [16][17][18][19][20]. It has been ascertained that, the convex functions played a very meaningful performance in the field of mathematical inequalities [2,[21][22][23][24]. ...
... Then clearly, the function Ψ is 4-convex with the given conditions. Therefore, putting Ψ(y) = y t r , q i = γ i and y i = ζ r i in (11), we receive (16). ...
... (ii) For the stated conditions, the function Ψ(y) = y t r is 4-convex. Therefore, to deduce (16) follow the procedure of (i). (iii) Obviously the function Ψ(y) = y t r is 4-concave for the aforementioned conditions. ...
In 2021, Ullah et al., introduced a new approach for the derivation of results for Jensen’s inequality. The purpose of this article, is to use the same technique and to derive improvements of Slater’s inequality. The planned improvements are demonstrated in both discrete as well as in integral versions. The quoted results allow us to provide relationships for the power means. Moreover, with the help of established results, we present some estimates for the Csiszár and Kullback–Leibler divergences, Shannon entropy, and Bhattacharyya coefficient. In addition, we discuss some additional applications of the main results for the Zipf–Mandelbrot entropy.
... Numerous mathematicians have generalized this inequality in numerous ways. For more related results, see [29][30][31][32][33][34][35][36][37][38][39][40][41][42][43][44]. The inequality is written as if T : I → R is a convex function on I and ı, ∈ I with ı ≤ such that: ...
... After adding (42) and (43), and integrating over [0, 1], we get ...
It is well known that both concepts of symmetry and convexity are directly connected. Similarly, in fuzzy theory, both ideas behave alike. It is important to note that real and interval-valued mappings are exceptional cases of fuzzy number-valued mappings (FNVMs) because fuzzy theory depends upon the unit interval that make a significant contribution to overcoming the issues that arise in the theory of interval analysis and fuzzy number theory. In this paper, the new class of p-convexity over up and down (UD) fuzzy relation has been introduced which is known as UD-p-convex fuzzy number-valued mappings (UD-p-convex FNVMs). We offer a thorough analysis of Hermite–Hadamard-type inequalities for FNVMs that are UD-p-convex using the fuzzy Aumann integral. Some previous results from the literature are expanded upon and broadly applied in our study. Additionally, we offer precise justifications for the key theorems that Kunt and İşcan first deduced in their article titled “Hermite–Hadamard–Fejer type inequalities for p-convex functions”. Some new and classical exceptional cases are also discussed. Finally, we illustrate our findings with well-defined examples.
... In recent decades, the power means have been applied very widely in several felds of science for the determination of multiple problems [43][44][45][46]. Te power means have a rich history, and a lot of literature is devoted to the power means [47,48]. ...
The field of mathematical inequalities has exerted a profound influence across a multitude of scientific disciplines, making it a captivating and expansive domain ripe for research investigation. This article offers estimations for the Slater difference through the application of the concept of convexity. We present a diverse type of applications that stem from the main findings related to power means, Zipf–Mandelbrot entropy, and within the field of information theory. Our main tools for deriving estimates for the Slater difference involve the triangular inequality, the definition of the convex function, and the well-established Jensen inequality.
... Recently, various inequalities have been given in the sense of generalizations and improvements for different fractional integrals. We state some of them here; the variants of Minkowski, Wirtinger, Hardy, Opial, Ostrowski, Hermite-Hadamard, Lyenger, Grüss, Cebyšev, and Pólya-Szegö [7][8][9][10][11][12][13][14][15]. Such applications of fractional integral operators compelled us to show the generalization of the reverse Minkowski inequality [7][8][9] involving general kernels. ...
In this article, we introduce a class of functions U(p) with integral representation defined over a measure space with σ-finite measure. The main purpose of this paper is to extend the Minkowski and related inequalities by considering general kernels. As a consequence of our general results, we connect our results with various variants for the fractional integrals operators. Such applications have wide use and importance in the field of applied sciences.
... The significance of power and quasi-arithmetic means have been fully accepted since 1930, and after that, a lot of researchers have given full attention to study their properties and characteristics in all manners. 53,54 Due to the great importance of these means many useful and interesting inequalities have accomplished while using different techniques and methods. 55,56 In the current section, we present various inequalities for power and quasi-arithmetic means. ...
The Jensen inequality is one of the most favorable inequalities during the last few decades due to its expressive characteristics and properties. This inequality has gained a very dominant position in the several fields of science as a result of its extensive applications. In the present note, we use an interesting approach for the determination of estimates for the Jensen difference. We acquire several estimates for the Jensen difference while utilizing the convex function definition, Jensen's inequality for concave functions, power mean, and Hölder inequalities. For the real visualizations of the acquired estimates, we provide some particular examples. We get different improvements for the Hölder inequality while taking specified functions in the received estimates. Moreover, we deduce some more results from the main work in the form of improvements for the Hermite‐Hadamard inequality. Furthermore, various relations for the famous quasi‐arithmetic and power means are acquired with the help of established estimates. At the end, we present various applications of the main work in information theory. The intended applications provide some new bounds for the Csiszár and Rényi divergences, Shannon entropy, and Bhattacharyya co‐efficient.
... Recently, various inequalities have been given in the sense of generalizations and improvements for different fractional integrals. We state some of them, that is, the variants of Minkowski, Wirtinger, Hardy, Opial, Ostrowski, Hermite-Hadamard, Lyenger, Grüss, Cebyšev and Pólya-Szegö [7,8,9,10,11,12,13,14,15]. Such applications of fractional integral operators compelled us to show the generalization of the reverse Minkowski inequality [7,8,9] involving general kernels. ...
In this article, we introduce a class of functions U(p) with integral representation defined over measure space with σ-finite measure. The main purpose of this paper is to extend the Minkowski and related inequalities by considering general kernels. As a consequence of our general results, we connect our results with various variants for the fractional integrals operators. Such applications have wide use and importance in the field of applied sciences.
AMS Subject Classification: 26D15; 26D10; 26A33; 34B27
... The significance of means has been fully accepted since 1930 and a number of researchers then gave full attention to the properties and applications of means [22,23]. A variety of article devoted to means in which they are studied very deeply in all directions. ...
The Jensen inequality has been reported as one of the most consequential inequalities that has a lot of applications in diverse fields of science. For this reason, the Jensen inequality has become one of the most discussed developmental inequalities in the current literature on mathematical inequalities. The main intention of this article is to find some novel bounds for the Jensen difference while using some classes of twice differentiable convex functions. We obtain the proposed bounds by utilizing the power mean and Höilder inequalities, the notion of convexity and the prominent Jensen inequality for concave function. We deduce several inequalities for power and quasi-arithmetic means as a consequence of main results. Furthermore, we also establish different improvements for Hölder inequality with the help of obtained results. Moreover, we present some applications of the main results in information theory.
... A real-valued function M : (0, ∞) × (0, ∞) → (0, ∞) is said to be a bivariate mean [3] if min{x, y} M(x, y) max{x, y} for all x, y ∈ (0, ∞). It is well-known that the bivariate means have wide applications in mathematics and other natural sciences [1,4,6,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,36,38], they have attracted the attention of many researchers [7,8,9,10,11,39,41,42,43,44,45,46,47,48,49,50,52,53,54,55,56,58,59,60]. Let x, y > 0 . ...
In the article, we prove that (Formula presented) are the best possible parameters on the interval [1/2,1] such that the double inequalities (Formula presented) hold for all u,v > 0 with u ≠ v and p ∈ [1/2,∞), where (Formula presented) are respectively the arithmetic, quadratic, contraharmonic, Seiffert and Neuman-S´andor means of u and v, and sinh⁻¹(x) = (Formula presented) is the inverse hyperbolic sine function.
... In the particular case p = x −1 and n = 2, M p,n (a, b) reduces to AG(a, b). Moreover, taking p = √ x and n = 1 , a new symmetric mean E(a, b) [17,40,41,67,96] involving the complete elliptic integral of the second kind was obtained ...
In the article, we present several new bounds for the the complete elliptic integrals K (r) = | 0π /²(1−r² sin² θ )⁻1/²dθ and E (r) = | 0π /² (1−r² sin² θ )¹/²dθ , and find an asymptotic expansion for K (r) as r → 1, which are the refinements and improvements of the previously well-known results.
... An enormous heft of present literature comprises generalizations of several variants by fractional integral operators and their applications [46][47][48][49][50][51][52]. We state some of them, that is, the variants of Minkowski, Hardy, Opial, Hermite-Hadamard, Grüss, Lyenger, Wrtinger, Ostrowski, Čebyšev and Pólya-Szegö [53][54][55][56][57][58][59]. Such applications of fractional integral operators compelled us to show the generalization of the reverse Minkowski inequality [43,44,53] involving generalized conformable fractional integrals operators. ...
This paper gives some novel generalizations by considering the generalized conformable fractional integrals operator for reverse Minkowski type and reverse Hölder type inequalities. Furthermore, novel consequences connected with this inequality, together with statements and confirmation of various variants for the advocated generalized conformable fractional integral operator, are elaborated. Moreover, our derived results are provided to show comparisons of convergence between old and modified operators towards a function under different parameters and conditions. The numerical approximations of our consequence have several utilities in applied sciences and fractional integro-differential equations.
... A real-valued function Ω : (0, ∞) × (0, ∞) → (0, ∞) is said to be a bivariate mean if min{ρ 1 , ρ 2 } ≤ Ω(ρ 1 , ρ 2 ) ≤ max{ρ 1 , ρ 2 } for all ρ 1 , ρ 2 ∈ (0, ∞). Recently, the properties and applications for the bivariate means and their related special functions have attracted the attention of many researchers [73][74][75][76][77][78][79][80][81][82][83][84][85][86]. In particular, many remarkable inequalities for the bivariate means can be found in the literature [87][88][89][90][91][92][93][94][95][96]. ...
The main aim of the present paper is to present a novel approach base on the exponentially convex function to broaden the utilization of celebrated Hermite-Hadamard type inequality. The proposed technique presents an auxiliary result of constructing the set of base functions and gives deformation equations in a simple form. The auxiliary result in the convexity has provided a convenient way of establishing the convergence region of several novel results. The strategy is not limited to the small parameter, such as in the classical method. The numerical examples obtained by the proposed approach indicate that the approach is easy to implement and computationally very attractive. The implementation of this numerical scheme clearly exhibits its effectiveness, reliability, and easiness regarding the applications in error estimates for weighted mean, the integral formula, rth moments of a continuous random variable, application to weighted special means and in developing the variants by extraordinary choices of n and θ as well as its better approximation.
In this study, we first propose some new concepts of coordinated up and down convex mappings with fuzzy-number values. Then, Hermite–Hadamard-type inequalities via coordinated up and down convex fuzzy-number-valued mapping (coordinated UD-convex FNVMs) are introduced. By taking the products of two coordinated UD-convex FNVMs, Pachpatte-type inequalities are also obtained. Some new conclusions are also derived by making particular decisions with the newly defined inequalities, and it is demonstrated that the recently discovered inequalities are expansions of comparable findings in the literature. It is important to note that the main outcomes are validated using nontrivial examples.
In this paper, with the use of newly defined class up and down log–convex fuzzy-number valued mappings, we offer a few new and original mappings defined by applying some mild restrictions over the definition of up and down log–convex fuzzy-number valued mapping. With the use of these mappings, we are able to develop partners of Fejér-type inequalities for up and down log–convexity, which improve upon certain previously established findings. The discussion also includes these mappings’ characteristics. Moreover, some nontrivial examples are also provided to prove the validation of our main results.
Using the power series expansion technique, this paper established two new inequalities for the sine function and tangent function bounded by the functions x2sin(λx)/(λx)α and x2tan(μx)/(μx)β. These results are better than the ones in the previous literature.
Using the power series expansions of the functions cotx,1/sinx and 1/sin2x, and the estimate of the ratio of two adjacent even-indexed Bernoulli numbers, we improve Cusa–Huygens inequality in two directions on 0,π/2. Our results are much better than those in the existing literature.
This work mainly investigates a class of convex interval-valued functions via the Katugampola fractional integral operator. By considering the p-convexity of the interval-valued functions, we establish some integral inequalities of the Hermite-Hadamard type and Hermite-Hadamard-Fejér type as well as some product inequalities via the Katugampola fractional integral operator. In addition, we compare our results with the results given in the literature. Applications of the main results are illustrated by using examples. These results may open a new avenue for modeling, optimization problems, and fuzzy interval-valued functions that involve both discrete and continuous variables at the same time. MSC: 26A51; 26D15; 05A30
In the article, we establish a sharp double inequality involving the ratio of generalized complete elliptic integrals of the first kind, which is the improvement and generalization of some previously known results.
For p ∈ R the p-th one-parameter mean Jp(a, b) of two positive numbers a and b with a ≠ b is defined by In this article, we answer the question: What are the greatest value α and the least value β, such that the double inequality Jα(a, b) < I(a, b) < Jβ(a, b) holds for all a, b > 0 with a ≠ b? Here denotes the identric mean of a and b.
In this paper, we find the largest value alpha and least value beta such that the double inequality L-alpha(a,b) < M(a,b) < L-beta(a,b) holds for all a,b > 0 with a not equal b. Here, M(a,b) and L-p(a,b) are the Neuman-Sandor and p-th generalized logarithmic means of a and b, respectively.
We present the best possible power mean bounds for the generalized logarithmic mean, and find the optimal lower generalized logarithmic mean bound for the identric mean.
In this paper, we find the greatest value alpha and the least values beta, p, q and r in (0,1/2) such that the inequalities L(alpha a + (1 - alpha)b, alpha b + (1 - alpha)a) < P(a, b) < L(beta a + (1 - beta)b, beta b + (1 - beta)a), H(pa + (1- p)b, pb + (1 - p)a) > G(a, b), H(qa + (1 - q)b, qb + (1 - q)a) > L(a, b), and G(ra + (1- r)b, rb + (1- r)a) > L(a, b) hold for all a, b > 0 with a not equal b. Here, H(a, b), G(a, b), L(a, b) and P(a, b) denote the harmonic, geometric, logarithmic and Seiffert means of two positive numbers a and b, respectively.
For , the power mean of order p, identric mean I(a,b) and arithmetic mean A(a,b) of two positive real numbers a and b are defined by
\begin{equation*}
M_{p}(a,b)=
\begin{cases}
\displaystyle\left(\tfrac{a^{p}+b^{p}}{2}\right)^{1/p}, & p\neq 0,\\
\sqrt{ab}, & p=0,
\end{cases}
\quad I(a,b)=
\begin{cases}
\displaystyle\tfrac{1}{\rm {e}}\left(b^{b}/a^{a}\right)^{1/(b-a)}, & a\neq b,\\
\displaystyle a, & a=b,
\end{cases}
\end{equation*}
and A(a,b)=(a+b)/2, respectively.
In the article, we answer the questions: What are the least values
p, q and r, such that inequalities
,
and
hold for all ?
We find the greatest value p and the least value q in ( 0,1 / 2 ) such that the double inequality H ( p a + ( 1 - p ) b , p b + ( 1 - p ) a ) < I ( a , b ) < H ( q a + ( 1 - q ) b , q b + ( 1 - q ) a ) holds for all a , b > 0 with a b . Here, H ( a , b ) , and I ( a , b ) denote the harmonic and identric means of two positive numbers a and b , respectively.
We answer the question: for any p , q ∈ ℝ with p q and p - q , what are the greatest value λ = λ ( p , q ) and the least value μ = μ ( p , q ) , such that the double inequality M λ ( a , b ) < M p ( a , b ) M q ( a , b ) < M μ ( a , b ) holds for all a , b > 0 with a b ? Where M p ( a , b ) is the p th power mean of two positive numbers a and b .
We present the best possible power mean bounds for the product Mpα(a,b)M-p1-α(a,b) for any p>0, α∈(0,1), and all a,b>0 with a≠b. Here, Mp(a,b) is the pth power mean of two positive numbers a and b.
In this paper, we present the greatest values , and p,
and the least values , and q such that the double inequalities
, and hold for all with
, where H(a,b)=2ab/(a+b), ,
, and
are the harmonic, Seiffert, quadratic, first
contraharmonic and second contraharmonic means of a and b, respectively.
We find the greatest value pp and least value qq in (0,1/2)(0,1/2) such that the double inequality G(pa+(1−p)b,pb+(1−p)a)<I(a,b)<G(qa+(1−q)b,qb+(1−q)a)G(pa+(1−p)b,pb+(1−p)a)<I(a,b)<G(qa+(1−q)b,qb+(1−q)a) holds for all a,b>0a,b>0 with a≠ba≠b. Here, G(a,b)G(a,b), and I(a,b)I(a,b) denote the geometric, and identric means of two positive numbers aa and bb, respectively.
Let ra, 0 <\alpha<\theta<\beta \lambda[M_n^{[\alpha]}]^{1-\lambda}[M_n^{[\beta]}]^{\lambda}\leq M_n^{[\theta]}(a)\theta\geq{1+ (\beta-\theta)/[m(\theta-\alpha)]}-1m = \min\{[2 + (n - 2)t^{\beta}]/[2 + (n - 2)t^{\alpha], t\in R_{++}\}; the smallest number, satisfying the reverse inequality, is. The method depends on the theory of majorization and analytic techniques. The optimality idea of establish-ing inequalities is displayed as many times as possible. As some applications, several inequalities (including improved Hardy's inequality) involving sums, integrals and ma-trices are obtained.
In a very interesting and recent note, Tung‐Po Lin [1] obtained the least value q and the greatest value p such that urn:x-wiley:01611712:media:ijmm718951:ijmm718951-math-0001 is valid for all distinct positive numbers x and y where urn:x-wiley:01611712:media:ijmm718951:ijmm718951-math-0002
The object of this paper is to give a simpler proof than Lin′s of a more general result. More precisely, the author obtained the classes of functions f α and h α , α ∈ R such that urn:x-wiley:01611712:media:ijmm718951:ijmm718951-math-0003
In 1992 Frank Hollad [1] stated the following inequality (1) (A 1 A 2 A n )<SUP>1/n</SUP> greater than or equal to 1/n (G 1 G 2 ...G n ) where A(k), G(k), k = 1, 2,..., n are arithmetic and geometric means, respectively, of positive numbers a(1), a(2),,...,a(k). In 1994 Kiran Kedlaya [2] gave a combinatorial proof of (1). In 1995 Takashi Matsuda [3] gave another proof of(1). In 1996 B. Mend and J. Pecaric [5] proved the following generalization of inequality (1) involving power means: (2) if S > T then, m(r,s) (a) greater than or equal to m(s,r) (a), where m(r,s) (a) is defined by the following definition (1.2). In this article a more general inequality, which concern weighted power means, is proved.
In this note several inqualities for generalized Heronian Means of two numbers are proved. These means are defined for the first time in this paper and they are compared with some well-known means. All inequalities are best possible.
In this paper we gave a generalization of power means which include positive nonlinear functionals. For these means we obtained generalizations of fundamental mean inequality, Hölder's and Minkowski's, and their converse inequalities.
A classical inequality for Euler’s gamma function states that
for all and with . We prove the following extension of this result. Let be the weighted power mean of of order r. The inequality
holds for all and with if and only if
We present various new inequalities involving the logarithmic mean L(x,y)=(x-y)/(logx-logy) L(x,y)=(x-y)/(\log{x}-\log{y}) , the identric mean I(x,y)=(1/e)(xx/yy)1/(x-y) I(x,y)=(1/e)(x^x/y^y)^{1/(x-y)} , and the classical arithmetic and geometric means, A(x,y)=(x+y)/2 A(x,y)=(x+y)/2 and G(x,y)=Ö{xy} G(x,y)=\sqrt{xy} . In particular, we prove the following conjecture, which was published in 1986 in this journal. If Mr(x,y) = (xr/2+yr/2)1/r(r ¹ 0) M_r(x,y)= (x^r/2+y^r/2)^{1/r}(r\neq{0}) denotes the power mean of order r, then M_c(x,y)<\frac{1}{2}(L(x,y)+I(x,y)) {(x,y>0,\, x\neq{y})} with the best possible parameter c=(log2)/(1+log2) c=(\log{2})/(1+\log{2}) .
Sharp inequalities are established between the Gaussian hypergeometric function and the power mean. These results extend known inequalities involving the complete elliptic integral and the hypergeometric mean.
In this paper, we generalize and sharpen the power means inequality by using the theory of majorization and the analytic techniques. Our results unify some optimal versions of the power means inequality. As application, a well-known conjectured inequality proposed by Janous et al. is proven. Furthermore, these results are used for studying a class of geometric inequalities for simplex, from which, some interesting inequalities including the refined Euler inequality and the reversed Finsler–Hadwiger type inequality are obtained.
The power mean and the logarithmic mean Generalization of the power means and their inequalities
- T P Lin
T. P. LIN, The power mean and the logarithmic mean, Amer. Math. Monthly, 81 (1974), 879–883.
[11] J. E. PĚ
CARI´CCARI´ CARI´C, Generalization of the power means and their inequalities, J. Math. Anal. Appl., 161
(1991), 395–404.