Abstract
In this paper we prove three power-exponential inequalities for positive real numbers. In particular, we conclude that this proofs give affirmatively answers to three, until now, open problems (Conjectures 4.4, 2.1 and 2.2) posed by Cîrtoaje (J. Inequal. Pure Appl. Math. 10:21, 2009; J. Nonlinear Sci. Appl. 4(2):130-137, 2011). Moreover, we present a new proof of the inequality for all positive real numbers a and b and . In addition, three new conjectures are presented.
Similar content being viewed by others
1 Introduction
The power-exponential functions have useful applications in mathematical analysis and in other theories like statistics [1], biology [2, 3], optimization [4], ordinary differential equations [5], and probability [6]. In recent years there has been intensive research in this area; see for instance [7–17] and the recent overview on general mathematical inequalities done by Cerone and Dragomir [18]. Some problems look like very simple but are difficult. For instance, we have the following two classical problems: find the solution of the equation and the basic problem of comparing and for all positive real numbers a and b. The first problem is perhaps one of the most ancient and useful problems concerning to power-exponential functions; see for instance [19–21]. It was introduced by Lambert in [22] and has been studied by recognized mathematicians like Euler, Pólya, Szegö, and Knuth; see [23–25]. The solution to the problem has inspired the definition of the well-known W-Lambert function; see [26]. For the solution to the second problem, see the discussion given in [27, 28] and more recently in [16]. Moreover, in spite of its algebraic simplicity, both problems are the central topic of a large number of research papers in the last years (see [7, 11, 13] and references therein). In particular, in this paper, we are interested in some inequalities conjectured by Cîrtoaje in [12, 29], which are very close to the second problem. To be more specific, we start by recalling that in [30] was introduced and probed the following assertion: the inequality holds for all positive real numbers less than or equal to 1. After that, Cîrtoaje [12] introduced, proved and conjectured several results about inequalities for power-exponential functions. In particular, in [12], it was established that the inequality
holds true for and for either or . However, in [12], Cîrtoaje leaves as an open problem the proof of (1.1) for . Moreover, in [12] the following conjectures were introduced:
Conjecture 4.3. If a, b, c are positive real numbers, then .
Conjecture 4.4. Let r be a positive real number. The inequality
holds true for all positive real numbers a, b, c with if and only if .
Conjecture 4.6. Let r be a positive real number. The inequality holds for all nonnegative real numbers a and b if and only if .
Conjecture 4.7. If a and b are nonnegative real numbers such that , then .
Conjecture 4.8. If a and b are nonnegative real numbers such that , then .
Afterwards, the analysis of (1.1) was completed by Manyama in [14]. Thereafter, of the Cîrtoaje conjectures, the milestones of the history are the works of Coronel and Huancas [13], Matejíčka [31], Li [9] and Hisasue [10] (see also the work of Cîrtoaje [29]), where they proved Conjectures 4.3, 4.6, 4.7, and 4.8, respectively. Here, we should be comment that the proof of Conjecture 4.4 is still open. Subsequently, in 2011 Cîrtoaje introduced a new proof of (1.1) and presented the following three new conjectures:
Conjecture 2.1. If and , then .
Conjecture 2.2. If , then .
Conjecture 5.1. If a, b are nonnegative real numbers satisfying , and if , then .
Recently, Miyagi and Nishizawa [7] have proved Conjecture 5.1. However, Conjectures 2.1 and 2.2 are still open. Thus, the main focus of this paper are the proofs of Conjectures 4.4, 2.1, and 2.2.
The main contribution of the present paper is the development of the proof of the following four theorems:
Theorem 1.1 The inequality (1.1) holds, for all positive real numbers a, b and for all .
Theorem 1.2 The inequality (1.2) holds, for all positive real numbers a, b, c and for all .
Theorem 1.3 The inequality
holds, for all positive real numbers a, b and for all .
Theorem 1.4 Let and . Then the inequality
holds.
Note that Conjectures 4.4, 2.1, and 2.2 are solved by Theorems 1.2, 1.3, 1.4, respectively. Moreover, we develop a proof of Theorem 1.1 which is an alternative proof of (1.1) for all positive real numbers a, b, and , which is distinct from the existing proofs given in [14, 29].
The rest of the paper is organized in two sections: In Section 2 we present the proofs of Theorems 1.1, 1.2, 1.3 and 1.4 and in Section 3 we present some remarks and three new conjectures.
2 Proofs of main results
In this section we present the proofs of Theorems 1.1, 1.2, 1.3, 1.4. Firstly, we recall a result of [13]. Then we present the corresponding proofs.
2.1 A preliminary result
For completeness and self-contained structure of the proofs of Theorems 1.1 and 1.2, we need the following result of [13].
Proposition 2.1 Consider with , and defined as follows:
Then the following properties are satisfied:
-
(i)
and .
-
(ii)
If , f is strictly increasing on and strictly decreasing on .
-
(iii)
If , f is strictly decreasing on and strictly increasing on .
-
(iv)
g is continuous on and strictly increasing on . Furthermore is a horizontal asymptote of .
2.2 Proof of Theorem 1.1
Without loss of generality, we assume that . Indeed, we find the proof (1.1) by application of Proposition 2.1 with , , and . Indeed, we distinguish three cases
(a2) Case ( and ). By Proposition 2.1(iv), we note that . Then, by the strictly increasing behavior of f (Proposition 2.1(ii)) we deduce the inequality since:
(b2) Case ( and ). For , we conclude the inequality by almost identical arguments to that used before in (i), since and . Otherwise, if , we deduce that
which implies the desired inequality.
(c2) Case ( and ). First, we define by the correspondence rule for and . The function h is concave and has a maximum at . Thus, we deduce that
Secondary, by the Napier inequality [32]
From (2.1) and (2.2) we have
which implies . The proof of this case is completed by application of Proposition 2.1(ii).
Hence, by (a2), (b2), and (c2) we conclude that Theorem 1.1 is valid.
2.3 Proof of Theorem 1.2
The proof of this theorem is again developed by application of Proposition 2.1. Firstly, we recall the notation of [13]:
The family is a set partition of . Now, with this notation, we subdivide the proof in three parts:
(a3) Case . This special case is a direct consequence of Theorem 1.1.
(b3) Case . If , we apply Theorem 1.1 and Proposition 2.1 as follows. We select , and , the monotonic behavior and properties of function f, defined on Proposition 2.1, imply that
since , and . Indeed, the corresponding proof of (2.3) needs the distinction of two cases: and . If , then and , so f is strictly increasing and implies (2.3). For , we note that and since and , then the assumption implies that and (2.3) is again true for this subcase. Moreover, for , by Theorem 1.1, we recall that the inequality
holds true for all . Adding (2.3) and (2.4) we deduce (1.2).
The proof for is similar to the case and we omit the details. However, we comment that for we choose , , and ; and for we select , , and .
(c3) Case . Without loss of generality, we assume that is such that , since the proof for is similar. We note that can be partitioned in the two sets
Now, we continue the proof by distinguish the following two subcases: and .
For the subcase , we apply the function f given on Proposition 2.1 with , , and to prove
Indeed, we firstly note that the function defined as follows: has the following properties:
(m a ) ;
(m b ) for all since ; and
(m c ) m has a maximum at , since the first and second derivatives of m are given by and and naturally and .
Moreover, we notice that is equivalent to , which is true for and ; see the proof of (2.1). Then, by (m a )-(m c ), it follows that , for all . In particular, for , we have
Now, from (2.6), we note that
which implies (2.5) by application of Proposition 2.1(ii), since and f is increasing on .
For the subcase , we apply the function f given on Proposition 2.1 with , , and to prove
We note that the inequality holds true for all . Now, in order to deduce that it is sufficient to prove that . Indeed, the function defined as has the following properties:
(q a ) ;
(q b ) for all , since ; and
(q c ) q is increasing in .
Then we deduce that for all . In particular, for , we deduce that , which implies the following sequence of implications:
Thus (2.8) holds true.
From (2.5) and (2.8), we deduce that
Hence, to complete the proof for , we add the inequality (2.9) with for , which is true by Theorem 1.1.
For we can follow line by line the proof of . However, we can obtain a direct proof by applying the result obtained for by interchanging the role of variables. For instance, if then , which implies (1.2).
Hence, by (a3), (b3), and (c3) we have the complete proof of Theorem 1.2.
2.4 Proof of Theorem 1.3
Given , we define the function as follows:
Then we prove that for all , which naturally implies the inequality for . Indeed, we prove that the function H has a global minimum at . The fact that in there is a local minimum of H follows by noticing that and , since
Meanwhile, the property that b is a global minimum of H can be proved by rewriting as the difference of two functions and by analyzing the sign of using some properties of this new functions. Indeed, to be more specific, we note that for all , where the functions K and Q are defined as follows:
The functions K and Q have the following properties:
(K1) K is strictly increasing on , since , for all .
(K2) when , , and .
(Q1) The derivative of Q is given by , for all . Then, in order to analyze the sign of , we introduce the set and a partition of Λ, where
We note that the sets , , are not empty since for instance for all , and . Moreover, we note that implies that and naturally is a subset of . The uniqueness of c can be deduced by noticing that the solution of is equivalent to the intersection of the following two monotone functions: and .
(Q2) when , and .
From (K1) and (Q1) we deduce the uniqueness of such that or equivalently . Now, from (K2) and (Q2), we note that for all since . Then for all . Additionally, from (K2) and (Q2), we observe that . This fact is a consequence of the fact that the function is strictly decreasing in r, since . Consequently, for we have for all . Hence, for we get or , which implies that for all . Thus, b is a global minimum of H. Therefore, for all and in particular for .
2.5 Proof of Theorem 1.4
The proof follows by the fact that the function is defined by the following correspondence rule:
and it has a global minimum at . Indeed, for simplicity of notation we develop the details of the proof for and with . Note that, in this case for an arbitrary , the function has the following form:
Then we have
An evaluation at implies that
Now, defining and , we observe that and . Then the Hessian matrix associated to P at is positive semidefinite since both functions, and , are positive on or equivalently the function P has a local minimum at . Now, we deduce that is the global minimum since we can prove that is the unique solution of . Indeed, assuming that there is with such that , we can deduce a contradiction. Note that
since the inequality holds for all and (see for instance [27]). Then , which is a contradiction with the assumption that . Thus, we see that is a global minimum of the function P or equivalently for all , which implies the desired inequality for .
3 Additional remarks on possible generalizations
In this section we present the possible extensions of Theorems 1.1, 1.2, and 1.3 to a sequence of positive real numbers. We note that the natural generalizations of (1.2) and (1.3) are given by
respectively. We present a partial proof of (3.1) (see Lemma 3.1, below) and leave as a conjecture the proof of (3.2).
Lemma 3.1 The inequality given in (3.1) holds for all , if we restrict to the hypercube .
Proof Before we start the proof, we notice that the function defined from and for is concave and . Then for all . Similarly, the function for is concave and for all . Now, we proceed by induction on n. Let us assume that the theorem is valid for a sequence of positive numbers for all . We note that
The terms and are positive by the inductive hypothesis. Meanwhile, the term is positive by the concavity of the functions ϒ and . Note that and , and or , depending if or , respectively. Then, by (3.3), it follows that the lemma is valid. □
Conjecture 3.1 Let and . Then the inequality (3.1) holds for all and .
Conjecture 3.2 Let and . Then the inequality (3.2) holds for all .
Conjecture 3.3 Let and . Then the inequality
holds for all .
References
Dahmani A, Karima Belaide K: Exponential inequalities in calibration problems with gaussians errors. Commun. Stat., Theory Methods 2013,42(19):3596-3607. 10.1080/03610926.2011.635257
Hou Q, Lin Z, Dusing RW, Gajewski BJ, Mccallum RW: A bayesian hierarchical assessment of gastric emptying with the linear, power exponential and modified power exponential models. Neurogastroenterology & Motility 2010,22(12):1308-1317. 10.1111/j.1365-2982.2010.01572.x
Floyd BNI, Camilleri M, Andresen V, Esfandyari T, Busciglio I, Zinsmeister AR: Comparison of mathematical methods for calculating colonic compliance in humans: power exponential, computer-based and manual linear interpolation models. Neurogastroenterology & Motility 2008,20(4):330-335. 10.1111/j.1365-2982.2007.01024.x
Park J-S, Baek J: Efficient computation of maximum likelihood estimators in a spatial linear model with power exponential covariogram. Comput. Geosci. 2001,27(1):1-7. 10.1016/S0098-3004(00)00016-9
Bruno AD: Power-exponential expansions of solutions to an ordinary differential equation. Dokl. Math. 2012,85(3):336-340. 10.1134/S106456241203009X
Fan X, Grama I, Liu Q: Large deviation exponential inequalities for supermartingales. Electron. Commun. Probab. 2012,17(59):1-8.
Miyagi M, Nishizawa Y: A short proof of an open inequality with power-exponential functions. Aust. J. Math. Anal. Appl. 2014. Article ID 6, 11(1): Article ID 6
Miyagi M, Nishizawa Y: Proof of an open inequality with double power-exponential functions. J. Inequal. Appl. 2013. Article ID 468, 2013: Article ID 468
Li Y: Solutions of two conjectures on inequalities with power-exponential functions. RGMIA Res. Rep. Collect. 2009. Article ID 7, 12(4): Article ID 7
Hisasue M: Solution of inequalities with power-exponential functions by Cîrtoaje. Aust. J. Math. Anal. Appl. 2012. Article ID 4, 9(2): Article ID 4
Cîrtoaje V: Proofs of three open inequalities with power-exponential functions. J. Nonlinear Sci. Appl. 2011,4(2):130-137.
Cîrtoaje V: On some inequalities with power-exponential functions. J. Inequal. Pure Appl. Math. 2009. Article ID 21, 10(1): Article ID 21
Coronel A, Huancas F:On the inequality . Aust. J. Math. Anal. Appl. 2012. Article ID 3, 9(1): Article ID 3
Manyama S: Solution of one conjecture on inequalities with power-exponential functions. Aust. J. Math. Anal. Appl. 2010. Article ID 1, 7(2): Article ID 1
Matejíčka L: On an open problem posed in the paper ‘Inequalities of power-exponential functions’. J. Inequal. Pure Appl. Math. 2008. Article ID 75, 9(3): Article ID 75
Qi F, Debnath L: Inequalities for power-exponential functions. J. Inequal. Pure Appl. Math. 2000. Article ID 15, 1(2): Article ID 15
Qi F, Xu S-L:The function : inequalities and properties. Proc. Am. Math. Soc. 1998,126(11):3355-3359. 10.1090/S0002-9939-98-04442-6
Cerone P, Dragomir SS: Mathematical Inequalities. CRC Press, Boca Raton; 2011.
Wright EM:Solution of the equation . Proc. R. Soc. Edinb., Sect. A 1959, 65: 193-203.
Wright EM:Solution of the equation . Bull. Am. Math. Soc. 1959, 65: 89-93. 10.1090/S0002-9904-1959-10290-1
Wright EM:Solution of the equation . Bull. Am. Math. Soc. 1960, 66: 277-281. 10.1090/S0002-9904-1960-10469-7
Lambert, JH: Observations variae in mathesin puram. Acta Helvitica, Physico-Mathematico-Anatomico-Botanico-Medica 63, 128-168 (1758)
Euler, J: De serie lambertina plurimisque eius insignibus proprietatibus. Acta Acad. Scient. Petropol. 2, 29-51 (1783)
Pólya G, Szegő G: Aufgaben und Lehrsätze aus der Analysis. Band II: Funktionentheorie, Nullstellen, Polynome Determinanten, Zahlentheorie. Springer, Berlin; 1971.
Corless RM, Gonnet GH, Hare DEG, Jeffrey DJ, Knuth DE: On the Lambert W function. Adv. Comput. Math. 1996,5(4):329-359.
Hoorfar A, Hassani M: Inequalities on the Lambert W function and hyperpower function. J. Inequal. Pure Appl. Math. 2008. Article ID 51, 9(2): Article ID 51
Bullen PS Pitman Monographs and Surveys in Pure and Applied Mathematics 97. In A Dictionary of Inequalities. Longman, Harlow; 1998.
Luo J, Wen JJ:A power-mean discriminance of comparing and . In Research Inequalities. Edited by: Yand X-Z. People’s Press of Tibet, The People’s Republic of China; 2000:83-88.
Cîrtoaje V: Proofs of three open inequalities with power-exponential functions. J. Nonlinear Sci. Appl. 2011,4(2):130-137.
zeikii, A, Cîrtoaje, V, Berndt, B:. Mathlinks Forum. http://www.mathlinks.ro/Forum/viewtopic.php?t=118722 (2006). Accessed November 2006
Matejíčka L: Solution of one conjecture on inequalities with power-exponential functions. J. Inequal. Pure Appl. Math. 2009,10(3):1-5.
Nelsen RB: Napier’s inequality (two proofs). Coll. Math. J. 1993. Article ID 165, 24(2): Article ID 165
Acknowledgements
We acknowledge the support of ‘Univesidad del Bío-Bío’ (Chile) through the research projects 124109 3/R, 104709 01 F/E, and 121909 GI/C.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Coronel, A., Huancas, F. The proof of three power-exponential inequalities. J Inequal Appl 2014, 509 (2014). https://doi.org/10.1186/1029-242X-2014-509
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-509