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A refinement of Sándor-Tóth's inequality
Journal of Inequalities and Applications volume 2012, Article number: 4 (2012)
Abstract
The objective of this article is to show a refinement of Sándor-Tóth's inequality related to the arithmetic functions which use unitary divisors. A new estimate of the average order of the arithmetic function given by Sándor-Tóth's inequality is suggested.
1 Introduction
The inequalities are important in many applications. Some attractive recent theoretical results related to inequalities include Jensen type inequalities [1], the general variational inequality problem [2], delay integral inequalities [3], inequalities that involve higher-order partial derivatives [4] or harmonic quasiconformal mappings [5]. We focus on inequalities that employ arithmetic functions based on the divisors of positive integers.
Among the divisors of a positive integer identifying a particular type of divisors, namely, the unitary divisors. But, first we present a brief history of this.
In [6], Vaidyanathaswamy introduced the notion of block-factor in the following way: a divisor d of n is a block-factor when , so the greatest common divisor of d and is 1. Later Cohen gave in [7] another terminology for block-factor which is currently referred to as unitary divisor.
For example, 4 is a unitary divisor of 12, because , but 2 is not a unitary divisor of 12, because .
We observe that for a prime power pa , the unitary divisors are 1 and pa .
Let τ*(n) denotes the number of unitary divisors of n, which is, in fact the number of the square free of n. Let denotes the sum of k th powers of the unitary divisors of n.
If is the prime factorization of n > 1, where p i are distinct primes and a i ≥ 1 for all i = 1, ..., r, then
and
where r is the number of distinct prime factors of n.
We note by γ(n) the largest divisor of n, which is squarefree, thus
and γ(1) = 1, by convention.
In [8, 9], Sándor and Tóth proved the inequality
for any n ≥ 1 and k ≥ 0.
This article aims two goals, a theoretical goal and an application goal. First, a refinement of this inequality is offered to fulfil the theoretical goal. Second, the Matlab mathematical software is used to analyze the behavior of the difference
in the case k = 1 and to fulfil the application goal.
Our new theoretical results are presented in the following section as a new inequality expressed as an improvement of (4). An application in terms of the Matlab-based solving of (5) is included as well. The conclusions are highlighted in Section 3.
2 Main results
Lemma 2.1. For any n ≥ 1 and k ≥ 0, the following inequality holds:
Proof. For n = 1, we obtain.
For n > 1 the canonical form of n is .
Using the inequality
which is true, for any prime number p for any a ≥ 0 and k ≥ 0.
Therefore, we derive the result
which implies the inequality
Consequently, the relation (6) is true.
□
We will find next, an expression of n for which the Sándor-Tóth inequality can be refined.
Theorem 2.2. For anywith a i ≥ 2 for all i = 1,..., n, the following inequality holds:
where k ≥ 1.
Proof. We first prove that
for , with a ≥ 2.
Since pa-1≥ pa/2, for any prime number p and for any a ≥ 2, it follows that
which is equivalent to
The combination of Lemma 2.1 and of the inequality (8) results finally in the inequality (7).
□
Remark 2.1. (a) If n is squarefree, then the relation
is true for any n ≥ 1 and k ≥ 0.
(b) The inequality (7) can be expressed in terms of
for any n ≥ 1 and k ≥ 0.
Lemma 2.3. For any n ≥ 1 and x i ≥ y i > 1, for all i = 1,..., n, we have
Proof. The mathematical induction is applied to prove this lemma. For n = 1, we obtain
which is true because it is equivalent to the inequality (x1 - 1)(y1 - 1) ≥ 0.
We consider that the inequality (11) is true for n and we will prove that it is also true for n + 1, thus:
Let us consider and , with x ≥ y. We will prove that
which is equivalent to the inequality
But xn+1- 1 ≥ yn+1- 1 ≥ 0, which means that the inequality (14) becomes, by minorization,
which is true.
The combination of the inequalities (12) and (13) leads to the result.
According to the principle of mathematical induction, the inequality (11) is true.
□
Another improvement of Sándor-Tóth's inequality is presented as follows in terms of Theorem 2.4.
Theorem 2.4. For any n ≥ 1 and k ≥ 1 there the following inequality holds:
Proof. The mathematical induction is also applied to prove this theorem. For n = 1, we have the equality in relation (15). If , then, from Lemma 2.3, we have
In fact, the inequality
is immediate because the arithmetic mean is greater than the geometric mean.
Let d be a divisor of n, then
so
The calculation of the sum for all divisors of n results in the relation
which is equivalent to the inequality
Therefore the proof is complete.
□
Corollary 2.5. For any n ≥ 1, the inequality
holds for any k ≥ 2.
Proof. Applying Theorem 2.4, we obtain
We apply the next inequality
which is equivalent to
and this is true for any n ≥ 1 and k ≥ 2.
Since the arithmetic mean is greater than the geometric mean, it follows that
Combining relations (17), (18), and (19), we derive the inequality (16).
□
Remark 2.2. The inequality (15) is an improvement of Sándor-Tóth's inequality, and we obtain the relation
for every n ≥ 1 and k ≥ 1.
Using the Matlab mathematical software we represent as follows the functions , Δ1(n) and in the same Cartesian coordinate system for n ≤ 10, 000, when Δ1(n) is a positive integer number (see Figure 1).
Theorem 2.6. For any n ≥ 1 and k ≥ 1, the inequality
holds, where σ k (n) is the sum of kth powers of the divisors of n and τ(n) is the number of divisors of n.
Proof. Using the identity of Dinghas [10], we prove the Radó inequality [11]
where A k is the arithmetic mean and G k is the geometric mean of k numbers of a1, a2,..., a n (k ≤ n).
Therefore, from the inequality (22), for n ≥ m, we derive the result
We consider that are the unitary divisors of n, and d1, d2,..., d s , ds+1,..., d t (t ≥ s) are all divisors of n, where . It follows, from the inequality (23), that
so
because
Consequently, the inequality (21) is proved.
□
Theorem 2.7. For n ≥ 1 and k ≥ 0, there is the inequality
Proof. Cartwright and Field proposed in [12] the inequality
Let 0 < m = min{x1, x2,..., x n } and let M = max{x1, x2,..., x n }.
Then
where .
If d1, d2,..., d s are the unitary divisors of n, we take and in inequality (25).
Therefore, we have m = 1, M = n and s = τ*(n), and the inequality (25) becomes:
Conducting simple calculations and accounting for
we observe that this inequality is equivalent to the inequality (24).
□
Remark 2.3. (a) The inequality (24) is another improvement of Sándor-Tóth's inequality. We also obtain the following result:
(b) Using the Matlab mathematical software we find the following characterization: if n is the square of an odd integer, then Δ k (n) is a positive integer. This fact proved relatively easily taking into account that
is a positive integer because p is odd prime number.
For example, if n ∈ {11025, 27225, 65029}, then Δ1(n) ∈ {1520, 3800, 9170}.
We find next an estimate of the average order of the function Δ1(n).
The average order of the function Δ1(n) is the sum
Theorem 2.8. For all x ≥ 1, we have
where ς is the Riemann zeta function, ς (3) is Apéry's constant with ς (3) = 1.2020569032... and O is the symbol of Landau.
Proof. Sándor and Kovács offered recently [13] a result related to the function τ(n), which is the number of divisors of n, namely,
for all n ≥ 1.
But, the number of divisors of n is greater than the number of unitary divisors of n, so
for all n ≥ 2.
Sitaramachandrarao and Surynarayana pointed out in [8] the following estimate of σ*(n):
Nathanson proved in [14] that if x and y are real numbers with y < [x], and f(t) if is a nonnegative monotonic function on [y, x], then
For , we find the average order of , thus
We will calculate the sum using the theorem of partial summation [14], thus
where f(n) and g(n) are two arithmetic functions, x ≥ 2, g(t) is continuously differentiable on [1, x], and .
Therefore, for f(n) = σ*(n), and (from (29)), relation (31) results in
so
Since , the application of (28) leads to
Summing up, the relations (29), (30), (32), and (33) lead to the fulfilment of (27).
3 Conclusions
This article has proposed a refinement of Sándor-Tóth's inequality, and two Matlab applications are given. Theorem 2.8 offers an approximation of the average order of Δ(x). Finding the average order of Δ(x) and the average order of
are subjects of future research. Studying the ideas above, we can identify other refinements of Sándor-Tóth's inequality.
The future research will also focus the extension of the area of applications of our new theoretical results. Such applications include solutions to optimal control problems [15], stability analysis [16, 17], robotics [18], fuzzy logic [19, 20], difference inequalities [21] or differential equations [22], as far as positive integers are concerned.
References
Agarwal RP, Dragomir SS: A survey of Jensen type inequalities for functions of selfadjoint operators in Hilbert spaces. Comput Math Appl 2010, 59(12):3785–3812. 10.1016/j.camwa.2010.04.014
Cho YJE, Petrot N: Regularization and iterative method for general variational inequality problem in Hilbert spaces. J Inequal Appl 2011, 2011: 21. 10.1186/1029-242X-2011-21
Feng Q, Meng F, Zhang Y, Zheng Bs, Zhou J: Some nonlinear delay integral inequalities on time scales arising in the theory of dynamics equations. J Inequal Appl 2011, 2011: 29. 10.1186/1029-242X-2011-29
Zhao C-J, Cheung W-S: On some Opial-type inequalities. J Inequal Appl 2011, 2011: 7. 10.1186/1029-242X-2011-7
Arsenovic M, Manojlovic V, Vuorinen M: Hölder continuity of harmonic quasiconformal mappings. J Inequal Appl 2011, 2011: 37. 10.1186/1029-242X-2011-37
Vaidyanathaswamy R: The theory of multiplicative arithmetic functions. Trans Am Math Soc 1931, 33: 579–662. 10.1090/S0002-9947-1931-1501607-1
Cohen E: Arithmetical functions associated with the unitary divisors of an integer. Math Z 1960, 74: 66–80. 10.1007/BF01180473
Sándor J, Crstici B: Handbook of Number Theory II. Springer, New York; 2006.
Sándor J, Tóth L: On certain number-theoretic inequalities. Fib Quart 1990, 28: 255–258.
Dinghas A: Some identities between arithmetic means and the other elementary symmetric functions of n numbers. Math Ann 1948, 120(1):154–157.
Hardy GH, Littlewood JE, Polya G: Inequalities. Cambrige University Press, Cambridge; 1952.
Cartwright DI, Field MJ: A refinement of the arithmetic mean-geometric mean inequality. Proc Am Math Soc 1978, 71(1):36–38. 10.1090/S0002-9939-1978-0476971-2
Sándor J, Kovács L: An inequality for the number of divisors of n . Octog Math Mag 2009, 17(2):746–749.
Nathanson M: Elementary Methods in Number Theory. Springer, New York; 2000.
Precup R-E, Preitl S, Rudas IJ, Tomescu ML, Tar JK: Design and experiments for a class of fuzzy controled servo systems. IEEE/ASME Trans Mechatron 2008, 13(1):22–35.
Pastravanu O, Matcovschi M-H: Diagonal stability of interval matrices and applications. Linear Algebra Appl 2010, 433(8–10):1646–1658. 10.1016/j.laa.2010.06.016
Park C, Jang SY, Lee JR, Yun D: On the stability of an AQCQ-functional equation in random normed spaces. J Inequal Appl 2011, 2011: 34. 10.1186/1029-242X-2011-34
Sekiyama K, Ito M, Fukuda T, Suzuki T, Yamashita K: An adaptive muscular force generation mechanism based on prior information of handling object. J Adv Comput Intell Intell Inf 2009, 13(3):222–229.
Mester G: Obstacle-slope avoidance and velocity control of wheeled mobile robots using fuzzy reasoning. In Proc 13th IEEE International Conference on Intelligent Engineering Systems (INES 2009). Barbados; 2009:245–249.
Sram N, Takacs M: Minnesota code: A neuro-fuzzy-based decision tuning. In Proc 15th IEEE International Conference on Intelligent Engineering Systems (INES 2011). Poprad, Slovakia; 2011:191–195.
Li D, Long S, Wang X: Difference inequality for stability of impulsive difference equations with distributed delays. J Inequal Appl 2011, 2011: 8. 10.1186/1029-242X-2011-8
Mir U, Merghem-Boulahi L, Gaïti D: A new framework for spectrum sharing based on multiagent systems and Petri nets. Int J Artif Intell 2011, 6(S11):144–160.
Acknowledgements
The authors were grateful to the referee for useful comments. This study was supported by the Romanian Ministry of Education, Research and Innovation through the PNII Idei project 842/2008.
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NM completed the main part of this article, CP presented the ideas of this article, REP participated in some study of this article. REP made the text file and communicated the manuscript. All authors read and approved the final manuscript.
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Minculete, N., Pozna, C. & Precup, RE. A refinement of Sándor-Tóth's inequality. J Inequal Appl 2012, 4 (2012). https://doi.org/10.1186/1029-242X-2012-4
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DOI: https://doi.org/10.1186/1029-242X-2012-4