- Research
- Open Access
- Published:
Power series inequalities via Young’s inequality with applications
Journal of Inequalities and Applications volume 2013, Article number: 314 (2013)
Abstract
In this paper, we establish some inequalities for power series with real coefficients by utilizing Young’s inequality for sequences of complex numbers. Some applications for special functions such as polylogarithm, hypergeometric and Bessel functions are also presented.
MSC:26D15.
1 Introduction
Let , , and let q satisfy . Then the classical Hölder’s inequality [[1], pp.19-21] states that
with equality holds if and only if the sequences and for are proportional and the is independent of k. The inequality (1.1) is reversed if .
The weighted version of Hölder’s inequality also holds, namely
where , , , , .
Tolsted in [2] (see also [[3], p.457], [[4], pp.63-64]) showed that Hölder’s inequality (also known in the literature as the Rogers inequality) can be easily proved by using Young’s inequality [5], namely
for any positive numbers x, y and with . Equality holds in (1.3) if and only if . For other applications and extensions of Young’s inequality, see [6, 7], [[8], pp.379-389] and references therein.
It is well known that Hölder’s inequality is one of the most important inequalities in real and complex analysis. For example, the celebrated Cauchy-Bunyakovsky-Schwarz (CBS) inequality (see [[9], p.16], [[8], p.83]) is a special case of Hölder’s inequality (1.1) for . Some other inequalities such as Minkowski’s inequality can be proved by using Hölder’s inequality.
Various extensions, generalizations, refinements, etc. of Hölder’s inequality have been obtained by several authors (see [10–20] and references therein). For instance, it comes to our attention that an interesting generalizations of Hölder’s inequality (1.1) by utilizing Young’s inequality (1.3), which was established by Dragomir and Sándor in [21] (see also [[22], pp.10-16]), is as follows:
for , , and with .
If now we consider an analytic function defined by the power series
with real coefficients and convergent on the disk , and apply the weighted version of Hölder’s inequality (1.2), then we can state that
for any with and is a new power series defined by , where with is the real signum function defined to be 1 if , −1 if and 0 if . The power series have the same radius of convergence as the original power series .
Motivated by the above results (1.6), (1.4) and the results from [21], and utilizing Young’s inequality, we established in this paper some inequalities for functions defined by power series with real coefficients. Particular examples that are related to some fundamental complex functions such as the exponential, logarithm, trigonometric and hyperbolic functions are presented. Applications for some special functions such as polylogarithm, hypergeometric and Bessel functions for the first kind are presented as well.
2 Some inequalities via Young’s inequality
On utilizing Young’s inequality (1.3) for power series with real coefficients, we establish the following result.
Theorem 1 Let and be two power series with real coefficients and convergent on the open disk , . If , and , so that , then
and
Proof If we choose , , in (1.3), then we have
for any .
Now, if we multiply this inequality (2.3) with positive quantities and summing over j and k from 0 to n, then we derive
Since all the series whose partial sums are involved in inequality (2.4) are convergent on the disk , taking the limit as in (2.4), we deduce the desired result (2.1).
Further, if we choose in (1.3), , , then we get
for any , .
Simplifying (2.5), we obtain that
for any .
Multiplying (2.6) by , and summing over j and k from 0 to n, we get
Since all the series whose partial sums are involved in inequality (2.7) are convergent on the disk , letting in (2.7), we deduce the desired inequality (2.2). □
The following particular case is of interest.
Corollary 1 If in (2.1) and (2.2), then
and
respectively, where , and with . In particular, if in (2.8) and (2.9), then we have
and
for any , with and is the complex signum function defined to be if and 0 if .
Remark 1 In the particular case in (2.8) and (2.9), we get the inequalities
and
respectively, for any with .
Some applications to particular functions of interest are as follows.
-
(1)
If we apply inequalities (2.8) and (2.9) to the function , , then we get
and
respectively, for any with , and , .
-
(2)
If we apply inequalities (2.8) and (2.9) to the function , , then we can state that
and
respectively, for any and with .
-
(3)
If we apply the function , , then from (2.8) and (2.9) we have
and
respectively, for any with , and , .
-
(IV)
Also, if we consider the function , , then, obviously, we have , . Applying inequalities (2.8) and (2.9) to this function, we get
and
respectively, for and , .
Similar results can be obtained for as well.
The following result also holds.
Theorem 2 Let and be as in Theorem 1. Then one has the inequalities
and
Proof If we choose in (1.3), , , , , we have
for any .
Multiplying (2.12) with and summing over j and k from 0 to n, we obtain that
From (1.3), we also have the inequality
for any , , , which was obtained by choosing , and repeating the same method as above.
Now, since all the series whose partial sums are involved in inequalities (2.13) and (2.14) are convergent on the disk , by letting in (2.13) and (2.14), respectively, we deduce the desired inequalities, i.e., (2.10) and (2.11). □
Corollary 2 If in (2.10) and (2.11), then
and
respectively, where , and with . In particular, if in (2.15) and (2.16), then we have
and
for with .
Remark 2 In the particular case in (2.15) or (2.16), we get the inequality
for any with .
In what follows, we provide some applications of inequalities (2.15) and (2.16) to particular functions of interest.
-
(1)
If we apply inequalities (2.15) and (2.16) to the function , , then we get
(2.17)
and
respectively, where , and with .
-
(2)
If we apply inequalities (2.15) and (2.16) to the function , , then we can state that
and
respectively, for any with .
-
(3)
If we take the function , , then from (2.15) and (2.16) we have
(2.18)
and
respectively, where , and with .
-
(4)
If we consider the function , , then we have , . Applying inequalities (2.15) and (2.16) to this function, we get
and
respectively, where , and with .
A similar result can be obtained for as well.
Theorem 3 Let and be as in Theorem 1. Then one has the inequalities
and
Proof Follows from inequality (1.3) on choosing , and , . That is, for any , we have the following inequalities:
and
respectively.
Repeating the same method as in Theorem 1 for (2.21) and (2.22), we deduce the desired inequalities, i.e., (2.19) and (2.20). □
As a particular case of interest, we can state the following corollary.
Corollary 3 If in (2.19) and (2.20), then
and
where , and with . In particular, if in (2.23) and (2.24), then we have
and
for , .
Inequalities (2.23) and (2.24) are also valuable sources of particular inequalities for complex numbers as will be outlined in the following.
-
(1)
If we apply inequalities (2.23) and (2.24) to the function , , then we get
and
respectively, for any , with .
-
(2)
If we apply inequalities (2.23) and (2.24) to the function , , then we can state that
and
respectively, for , .
-
(3)
If we apply the function , , then from (2.23) and (2.24) we have
and
respectively, for any , with .
-
(4)
If we consider the function , , then we have , . Applying inequalities (2.23) and (2.24) to this function, we get
and
respectively, for any , .
A similar result can be obtained for as well.
3 Applications to special functions
In this section, we give some inequalities for some special functions such as polylogarithm, hypergeometric, Bessel and modified Bessel functions for the first kind. Before that, we state here some basic concepts and definitions of those functions.
The polylogarithm is a function defined by the power series
which converges absolutely for all complex values of the order n and the argument z where . It is also known in the literature as Jonquiére’s function. The special cases reduce to and , where ζ and η are the Riemann zeta function and Dirichlet eta function, respectively. When , the first polylogarithm involves the ordinary logarithm, i.e., , while the second
is called the dilogarithm or Spence’s function.
For other integer values of order n, the polylogarithm reduces to the ratio of a polynomial in z, for instance,
The hypergeometric function is defined for all by the series
where with and the , is a Pochhammer symbol which is defined by
Hypergeometric function (3.3) with particular arguments of a, b and c reduces to elementary functions. For instance,
Further, the Bessel functions of the first kind, denoted as , are defined by the power series
for with . If z is replaced by arguments , then from (3.4) we have
for with . These functions (3.5) are called the modified Bessel functions of the first kind.
It is clearly seen that from (3.1), (3.3), (3.4) and (3.5), that is, , , and are power series with real coefficients and convergent on the open disk . Therefore, all the results in the above section hold true. For instance, from (2.15) we have the following corollaries.
Corollary 4 If is the polylogarithm function, then we have
for any , with and , .
In particular, if in (3.6), then we have the following inequality:
for all , and with .
If we take in (3.6), then we get inequality (2.18) for all with and , .
Also, if we choose in (3.6) , then we obtain
for any , and with . is the dilogarithm function which is defined in (3.2).
Corollary 5 If is a hypergeometric function, then for any , we have
where with and , .
In particular, if we choose , in (3.7), then we get inequality (2.17). Also, if we choose , , then inequality (3.7) reduces to (2.18).
Corollary 6 If and are the Bessel and modified Bessel functions of the first kind, respectively, then for any , we have
where , and with .
In particular, if in (3.8), then for with , we get
where .
Other inequalities involving the polylogarithm, hypergeometric, Bessel and modified Bessel functions can be found in the literature (see [23–28] and references therein).
References
Beckenbach EF, Bellman R: Inequalities. Springer, Berlin; 1961.
Tolsted E: An elementary derivation of the Cauchy, Hö lder and Minkowski inequalities from Young’s inequality. Math. Mag. 1964, 37: 2–12. 10.2307/2688239
Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications. Academic Press, New York/London; 1979.
Rudin W: Real and Complex Analysis. 3rd edition. McGraw-Hill, New York; 1987.
Young WH: On classes of summable functions and their Fourier series. Proc. R. Soc. Lond. A 1912, 87(594):225–229. 10.1098/rspa.1912.0076
Cerone P: On Young’s inequality and its reverse for bounding the Lorenz curve and Gini mean. J. Math. Inequal. 2009, 3(3):369–381.
Hong FH, Yeh CC, Yu SL, Hong CH: Young’s inequality and related results on time scales. Appl. Math. Lett. 2005, 18: 983–988. 10.1016/j.aml.2004.06.028
Mitrinovic DS, Pečaric JE, Fink AM: Classical and New Inequalities in Analysis. Kluwer Academic, Dordrecht; 1993.
Hardy G, Littlewood JE, Pólya G: Inequalities. Cambridge University Press, Cambridge; 1952.
Abramovich S, Mond B, Pecaric JE: Sharpening Hölder’s inequality. J. Math. Anal. Appl. 1995, 196: 1131–1134. 10.1006/jmaa.1995.1465
Carroll JA, Cordner R, Evelyn CJA: A new extension of Hölder’s inequality. Enseign. Math. 1970, 16: 69–71.
Daykin DE, Eliezer CJ: Generalization of Hölder’s and Minkowski’s inequalities. Proc. Camb. Philos. Soc. 1968, 64: 1023–1027. 10.1017/S0305004100043747
He WS: Generalization of a sharp Hölder’s inequality and its application. J. Math. Anal. Appl. 2007, 332: 741–750. 10.1016/j.jmaa.2006.10.019
Kim YI, Yang X: Generalizations and refinements of Hölder’s inequality. Appl. Math. Lett. 2012, 25: 1094–1097. 10.1016/j.aml.2012.03.027
Mitrinovic DS, Pečaric JE: On an extension of Hölder’s inequality. Boll. Un. Mat. Ital. A (7) 1990, 4: 405–408.
Kwon EG: Extension of Hölder’s inequality (1). Bull. Aust. Math. Soc. 1995, 51: 369–375. 10.1017/S0004972700014192
Qiang H, Hu Z: Generalizations of Hölder’s and some related inequalities. Comput. Math. Appl. 2011, 61: 392–396. 10.1016/j.camwa.2010.11.015
Yang X: Hölder’s inequality. Appl. Math. Lett. 2003, 16: 897–903. 10.1016/S0893-9659(03)90014-0
Yang X: A generalization of Hölder’s inequality. J. Math. Anal. Appl. 2000, 247: 328–330. 10.1006/jmaa.2000.6873
Yang X: Refinement of Hölder’s inequality and application to Ostrowski inequality. Appl. Math. Comput. 2003, 138: 455–461. 10.1016/S0096-3003(02)00159-5
Dragomir SS, Sándor J: Some generalizations of Cauchy-Bunyakovsky-Schwarz’s inequality. Gaz. Mat. Metod. (Bucharest) 1990, 11: 104–109. (in Romanian)
Dragomir SS: Discrete Inequalities of the Cauchy-Bunyakovsky-Schwarz Type. Nova Publ., New York; 2004.
Baricz A: Functional inequalities involving Bessel and modified Bessel functions of the first kind. Expo. Math. 2008, 26: 279–293. 10.1016/j.exmath.2008.01.001
Barnard RW, Kendall KC: On inequalities for hypergeometric analogues of the arithmetic-geometric mean. J. Inequal. Pure Appl. Math. 2007, 8(3):1–12.
He B, Yang B: On a Hilbert-type inequality with a hypergeometric function. Commun. Math. Anal. 2010, 9(1):84–92.
Jemai MM: A main inequality for several special functions. Comput. Math. Appl. 2010, 60: 1280–1289. 10.1016/j.camwa.2010.06.007
Yadava SR, Singh B: Certain inequalities involving special functions. Proc. Natl. Acad. Sci. India, Sect. a Phys. Sci. 1987, 57(3):324–328.
Zhu L: Jordan type inequalities involving the Bessel and modified Bessel functions. Comput. Math. Appl. 2010, 59: 724–736. 10.1016/j.camwa.2009.10.020
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The first author AI is currently a PhD student under supervision of the second author SSD and the third author MD is the co-supervisor. They jointly worked on deriving the results. 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 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Ibrahim, A., Dragomir, S.S. & Darus, M. Power series inequalities via Young’s inequality with applications. J Inequal Appl 2013, 314 (2013). https://doi.org/10.1186/1029-242X-2013-314
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2013-314
Keywords
- Young’s inequality
- Hölder’s inequality
- power series