Matejíˇcka Journal of Inequalities and Applications (2017) 2017:159 DOI 10.1186/s13660-017-1436-6

RESEARCH

Open Access

Next generalization of Cîrtoaje’s inequality Ladislav Matejíˇcka* Dedicated to my wife Emília Matejíˇcková *

Correspondence: [email protected] Faculty of Industrial Technologies in Púchov, Trenˇcín University of Alexander Dubˇcek in Trenˇcín, I. Krasku 491/30, Púchov, 02001, Slovakia

Abstract In this paper, we classify sets of solutions of the next generalized Cîrtoaje’s inequality and its reverse, respectively. MSC: 26D15 Keywords: inequalities with power-exponential functions; Cîrtoaje’s inequality

1 Introduction In recent years, inequalities with power-exponential functions have been intensively studied [–]. They have many important applications. For example, they can be found in mathematical analysis and in other theories like mathematical physics, mathematical biology, ordinary differential equations, probability theory and statistics, chemistry, economics. For more details, a literature review and the history of inequalities with powerexponential functions, see []. Cîrtoaje, in [], has introduced the following interesting conjecture on the inequalities with power-exponential functions. The inequality is similar to the reverse arithmetic-geometric mean inequality where its terms were rearranged. Conjecture  If a, b ∈ (, ] and r ∈ (, e], then √  ara brb ≥ arb + bra .

()

The conjecture was proved by Matejíčka []. Matejíčka also proved () under other conditions in [, ]. For example, he showed that () is also valid for a, b, r ∈ (, e]. In [], one interesting property of the generalized Cîrtoaje’s inequality (CI) was found. In [], a classification of sets of solutions of (CI)   n n–   rx rx n  + xi i ≥ xrx xi i+ n n i=

()

i=

was made.

2 Methods In this paper, methods of mathematical and numerical analysis are used. We make a classification of sets of solutions of the other generalization of (CI). © The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided 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.

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Let ϕ, ψ be functions from {, . . . , n} to {, . . . , n}, where n ∈ N . Put  n   n   r  rxψ(i) ln xi . F(r) = ln n + xϕ(i) ln xi – ln e n i= i=

()

The function F(r) is defined on Rn+ where n ∈ N, r ≥ , Rn+ = {(x , . . . , xn ), xi > , i = , . . . , n}. We note that F(r) ≥  is equivalent to the following generalization of Cîrtoaje’s inequality (I):   n n  rxϕ(i)  rx n xi ≥ xi ψ(i) . n i=

()

i=

The reverse inequality to (I)   n n  rxϕ(i)  rx n n xi < xi ψ(i) i=

()

i=

we denote by (RI).

3 Results and discussion We remark that in [] the special case of our classification for () was presented, where ϕ(i) = i, ψ(i) = i + , i = , . . . , n – , ϕ(n) = n, ψ(n) = . The following functions:  xϕ(i) log(xi ) – mx , n i= n

g(x , . . . , xn ) =

()



where mx = max xψ(m) log(xm ) , ≤m≤n

h(x , . . . , xn ) =

n  (xϕ(i) – xψ(i) ) log(xi ),

()

i=

we will call characteristic functions of (I). Put

Sn = (x , . . . , xn ) ∈ Rn+ ; xi = xj , i, j = , . . . , n . We prove the following lemma. Lemma  Let F(r) be defined by (). Let ϕ, ψ be arbitrary functions from {, . . . , n} to {, . . . , n}, n ∈ N . Then F(r) is a concave function for each A ∈ Rn+ – Sn , and F() = . If there is i = j; i, j ∈ N such that xψ(i) ln xi = xψ(j) ln xj , then F(r) is a strongly concave function in A. Proof F() =  is evident. Easy calculation gives    n n erxψ(i) ln xi xψ(i) ln xi   xϕ(i) ln xi – i=n rx ln x F (r) = i ψ(i) n i= i= e 

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and –L(r) F  (r) = n ( i= exp(rxψ(i) ln xi )) where L(r) =

n  n 

exp(rxψ(i) ln xi + rxψ(j) ln xj )xψ(i) ln xi

i= j=

–

n 

exp(rxψ(i) ln xi )

i=

n 

exp(rxψ(j) ln xi )(xψ(i) ln xi )xψ(j) ln xj

j=

  = exp(rxψ(i) ln xi + rxψ(j) ln xj )xψ(i) ln xi  i= j= n

n

  exp(rxψ(i) ln xi + rxψ(j) ln xj )xψ(j) ln xj  i= j= n

+

–

n 

n

exp(rxψ(i) ln xi )

i=

=

n  n 

n 

exp(rxψ(j) ln xi )(xψ(i) ln xi )xψ(j) ln xj

j=

exp(rxψ(i) ln xi + rxψ(j) ln xj )(xψ(i) ln xi – xψ(j) ln xj ) ≥ .

i= j=

The proof is completed.



Now we prove the following lemma. Lemma  Let g, h be defined by (), (). Let ϕ, ψ be arbitrary functions from {, . . . , n} to {, . . . , n}, n ∈ N . Then there are five cases.  . If h(x , . . . , xn ) = ni= (xϕ(i) – xψ(i) ) log(xi ) <  for A = (x , . . . , xn ) ∈ Rn+ then (RI) is valid for all r >  in A = (x , . . . , xn ) ∈ Rn+ .  . If h(x , . . . , xn ) = ni= (xϕ(i) – xψ(i) ) log(xi ) =  and  g(x , . . . , xn ) = n ni= xϕ(i) log(xi ) – max≤m≤n {xψ(m) log(xm )} <  for A = (x , . . . , xn ) ∈ Rn+ then (RI) is valid for all r >  in A = (x , . . . , xn ) ∈ Rn+ .  . If h(x , . . . , xn ) = ni= (xϕ(i) – xψ(i) ) log(xi ) =  and  g(x , . . . , xn ) = n ni= xϕ(i) log(xi ) – max≤m≤n {xψ(m) log(xm )} =  for A = (x , . . . , xn ) ∈ Rn+ then F(r) =  for r ≥  in A = (x , . . . , xn ) ∈ Rn+ .  . If h(x , . . . , xn ) = ni= (xϕ(i) – xψ(i) ) log(xi ) >  and  g(x , . . . , xn ) = n ni= xϕ(i) log(xi ) – max≤m≤n {xψ(m) log(xm )} ≥  for A = (x , . . . , xn ) ∈ Rn+ then (I) is valid for all r ≥  in A = (x , . . . , xn ) ∈ Rn+ .  . If h(x , . . . , xn ) = ni= (xϕ(i) – xψ(i) ) log(xi ) >  and  g(x , . . . , xn ) = n ni= xϕ(i) log(xi ) – max≤m≤n {xψ(m) log(xm )} <  for A = (x , . . . , xn ) ∈ Rn+ then there is r >  such that (I)is valid for r ∈ (, r ] and (RI) is valid for r ∈ (r , ∞) in A = (x , . . . , xn ) ∈ Rn+ . Proof The proof is evident. It follows from Lemma .



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Note  It is easy to see that if g(x , . . . , xn ) =  and h(x , . . . , xn ) =  then F(r) =  for all r ≥ . Really, from Lemma  we have F  () =  and limr→∞ F  (r) = . If F(r ) =  for some r >  then F(r ) < . Then there exists z such that F(r ) – F() = F  (z)r and  < z < r . It implies F  (z) < . Because of F  (r) ≤  we get F  is non-increasing for r ≥ . For r > z >  we obtain F  (r) ≤ F  (z) so limr→∞ F  (r) ≤ F  (z) < . This is a contradiction.

4 Conclusion In this paper, we showed the following. If (I) is valid in (x , . . . , xn ) for some r >  then (I) is valid in (x , . . . , xn ) for all  < r ≤ r . Similarly, if (RI) is valid in (x , . . . , xn ) for some r >  then (RI) is valid in (x , . . . , xn ) for all r > r . We think that the way how to classify sets of solutions of the power-exponential inequalities could be used for other suitable inequalities. Now we give examples of concrete applications of our results. We make the complete classification of sets of solutions for (I) and (RI) inequalities where n = . Using Matlab for plotting graphs of the solution curves for the characteristic equations g(X) = , h(X) =  we obtain the following figures for (I) and (RI). In the figures we denote by I + RI the points where (I) and also (RI) are locally valid. By I we denote points where (I) is valid for all r >  and by RI we denote points where (RI) is valid for all r > . It is easy to show that for n =  there are only  basic cases of inequalities (I). The other four cases of (I) can be transformed to the previous cases. Example  Let n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we have (I): rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x ) = ,



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) . See Figure .

Figure 1 Solution points for inequalities Examples 1, 7, 8, 11.

()

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Figure 2 Solution points for inequalities Example 2.

Figure 3 Solution points for inequalities Example 3.

Example  Let us consider that n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then by (I) rx rx  rx  xrx  x ≥ x + x ,

     x , h(x , x ) = (x – x ) log(x ) + (x – x ) log(x ) = (x – x ) ln    x



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) .

()

See Figure . Example  Put n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we obtain by (I) rx rx  rx  xrx  x ≥ x + x ,   h(x , x ) = (x – x ) log(x ) + (x – x ) log(x ),  



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) . See Figure .

()

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Figure 4 Solution points for inequalities Example 4.

Figure 5 Solution points for inequalities Example 5.

Example  Let us consider n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we get by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x )

()

= (/)(x – x ) log(x ),



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) . See Figure . Example  Let n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we have by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x ) = (/)(x – x ) log(x x ),



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) . See Figure .

()

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Figure 6 Solution points for inequalities Example 6.

Example  Put n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x )   x , = (/)(x – x ) log x



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) .

()

See Figure . Example  Let us consider n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we obtain by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x ) = ,



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) .

()

See Figure . Example  Put n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x ) = ,



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) . See Figure .

()

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Figure 7 Solution points for inequalities Example 9.

Figure 8 Solution points for inequalities Example 10.

Example  Let us consider n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we get by (I) rx rx  rx  xrx  x ≥ x + x ,    h(x , x ) = (x – x ) log(x ) + (x – x ) log(x ) = (x – x ) ln x ,   



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) .

()

See Figure . Example  Let n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then by (I) rx rx  rx  xrx  x ≥ x + x ,    h(x , x ) = (x – x ) log(x ) + (x – x ) log(x ) = (x – x ) log(x ),   



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) . See Figure .

()

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Figure 9 Solution points for inequalities Example 12.

Example  Put n = , ϕ() = , ϕ() = , ψ() = , ψ() = . We obtain by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x ) = ,



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) .

()

See Figure . Example  Let n = , ϕ() = , ϕ() = , ψ() = , ψ() = . Then we have by (I) rx rx  rx  xrx  x ≥ x + x , h(x , x ) = (/)(x – x ) log(x ) + (/)(x – x ) log(x ) = (/)(x – x ) log(x ),



g(x , x ) = (/) x log(x ) + x log(x ) – max x log(x ), x log(x ) .

()

See Figure .

Acknowledgements The work was supported by VEGA grant no. 1/0649/17 and KEGA grant no. 007TnUAD-4/2017. The author thanks Professor Ondrušová, Dean of the faculty FPT TnUAD in Púchov, Professor Vavro and Ing. Balážová for their kind support and he is deeply grateful to the unknown reviewer for valuable remarks and suggestions. Competing interests The author declares that he has no competing interests. Author’s contributions All authors read and approved the final manuscript.

Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 30 April 2017 Accepted: 5 June 2017 References 1. Cîrtoaje, V: Proofs of three open inequalities with power exponential functions. J. Nonlinear Sci. Appl. 4(2), 130-137 (2011) 2. Coronel, A, Huancas, F: The proof of three power exponential inequalities. J. Inequal. Appl. 2014, 509 (2014). http://www.journalofinequalitiesandapplications.com/content/2014/1/509 3. Matejíˇcka, L: Proof of one open inequality. J. Nonlinear Sci. Appl. 7, 51-62 (2014). http://www.tjnsa.com 4. Matejíˇcka, L: On the Cîrtoaje’s conjecture. J. Inequal. Appl. 2016, 152 (2016). doi:10.1186/s13660-016-1092-2

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5. Matejíˇcka, L: Some remarks on Cîrtoaje’s conjecture. J. Inequal. Appl. 2016, 269 (2016). doi:10.1186/s13660-016-1211-0 6. Matejíˇcka, L: On one interesting inequality. Surv. Math. Appl. 11, 1842-6298 (2016) 7. Miyagi, M, Nishizawa, Y: A stronger inequality of Cîrtoaje’s one with power exponential functions. J. Nonlinear Sci. Appl. 8, 224-230 (2015). http://www.tjnsa.com