Complete description of the monotonicity of a class of sequences with a parameter

Stevo Stević; Stevo Stević

doi:10.3934/math.2026838

AIMS Mathematics

2026, Volume 11, Issue 7: 20606-20618. doi: 10.3934/math.2026838

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Research article

Complete description of the monotonicity of a class of sequences with a parameter

Stevo Stević ^{1,2,3
,
,}

1.
Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/Ⅲ, 11000 Beograd, Serbia
2.
Serbian Academy of Sciences and Arts, Knez Mihailova 35, 11000 Beograd, Serbia
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan

Received: 23 March 2026 Revised: 11 June 2026 Accepted: 12 June 2026 Published: 13 July 2026
MSC : 40A05, 52A41

A complete description of the monotonicity character of the convex combinations of the sequences
$ x_n = \sum\limits_{j = 1}^n\frac1{j^{\alpha}}-\frac{n^{1-{\alpha}}}{1-{\alpha}}{\quad\mbox{ and }\quad} y_n = \sum\limits_{j = 1}^n\frac1{j^{\alpha}}-\frac{(n+1)^{1-{\alpha}}}{1-{\alpha}}, \quad n\in {\mathbb N}, $
when $ {\alpha}\in(0, 1), $ on the whole domain $ {\mathbb N} $, is given, considerably extending some special cases in the literature, and solving a highly nontrivial open problem completely. An important part in the study plays a few analytic inequalities that have been proved and used in the proof of the result that characterizes the monotonicity character of the sequences.
- real sequences,
- monotonicity character of sequences,
- convex combinations
Citation: Stevo Stević. Complete description of the monotonicity of a class of sequences with a parameter[J]. AIMS Mathematics, 2026, 11(7): 20606-20618. doi: 10.3934/math.2026838

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Abstract

A complete description of the monotonicity character of the convex combinations of the sequences

$ x_n = \sum\limits_{j = 1}^n\frac1{j^{\alpha}}-\frac{n^{1-{\alpha}}}{1-{\alpha}}{\quad\mbox{ and }\quad} y_n = \sum\limits_{j = 1}^n\frac1{j^{\alpha}}-\frac{(n+1)^{1-{\alpha}}}{1-{\alpha}}, \quad n\in {\mathbb N}, $

when $ {\alpha}\in(0, 1), $ on the whole domain $ {\mathbb N} $, is given, considerably extending some special cases in the literature, and solving a highly nontrivial open problem completely. An important part in the study plays a few analytic inequalities that have been proved and used in the proof of the result that characterizes the monotonicity character of the sequences.

References

[1]	D. Adamović, Collected works, Belgrade: University of Belgrade, Faculty of Mathematics, 2009.
[2]	M. I. Bashmakov, B. M. Bekker, V. M. Gol'hovoi, Zadachi po matematike. Algebra i analiz, (in Russian), Nauka, 1982.
[3]	L. Berg, On the asymptotics of nonlinear difference equations, Z. Anal. Anwend., 21 (2002), 1061–1074. https://doi.org/10.4171/zaa/1127 doi: 10.4171/zaa/1127
[4]	A. Y. Dorogovtsev, Matematika segodnya '90, (in Russian), Kiev: Vishcha Shkola, 1990.
[5]	L. Eulero, De progressionibvs harmonicis observationes, Comm. Acad. Sci. Petrop., 7 (1740), 150–161.
[6]	S. Golubev, R. Ushakov, Reshenie zadachi 485, Kvant, 11 (1978), 20–22.
[7]	J. Hadamard, Étude sur les propriétés des fonctions entiéres et en particulier d'une fonction considérée par Riemann, J. Math. Pures Appl., 58 (1893), 171–215.
[8]	C. Hermite, Sur deux limites d'une intégrale définie, Mathesis, 3 (1883), 6.
[9]	B. D. Iričanin, Dynamics of a class of higher order difference equations, Discrete Dyn. Nat. Soc., 2007 (2007), 073849.
[10]	C. M. Kent, Convergence of solutions in a nonhyperbolic case, Nonlinear Anal., 47 (2001), 4651–4665. https://doi.org/10.1016/S0362-546X(01)00578-8 doi: 10.1016/S0362-546X(01)00578-8
[11]	P. P. Korovkin, Inequalities, Moscow: Mir Publishers, 1975.
[12]	W. A. J. Kosmala, A friendly introduction to analysis: Single and multivariable, 2 Eds., Pearson Prentice Hall, 2004.
[13]	V. A. Krechmar, A problem book in algebra, Moscow: Mir Publishers, 1978.
[14]	I. I. Lyashko, A. K. Boyarchuk, Y. G. Gay, G. P. Golovach, Matematicheskiy analiz v primerah i zadachah, (in Russian), Vishcha Shkola, 1974.
[15]	T. Martynenko, R. Ushakov, Zadacha 485, Kvant, 1 (1978), 28–29.
[16]	D. S. Mitrinović, Analytic inequalities, Berlin: Springer-Verlag, 1970. https://doi.org/10.1007/978-3-642-99970-3
[17]	D. S. Mitrinović, Matematička indukcija, binomna formula, kombinatorika, (in Serbian), Beograd: Gradjevinska Knjiga, 1980.
[18]	D. S. Mitrinović, D. D. Adamović, Nizovi i redovi: Definicije, stavovi, zadaci, problemi, (in Serbian), Beograd: Naučna Knjiga, 1980.
[19]	D. S. Mitrinović, J. D. Kečkić, Metodi izračunavanja konačnih zbirova, (in Serbian), Beograd: Naučna Knjiga, 1984.
[20]	N. Nielsen, Handbuch der theorie der gammafunktion, (in German), Teubner, 1906.
[21]	G. Papaschinopoulos, C. J. Schinas, On a system of two nonlinear difference equations, J. Math. Anal. Appl., 219 (1998), 415–426. https://doi.org/10.1006/jmaa.1997.5829 doi: 10.1006/jmaa.1997.5829
[22]	G. Papaschinopoulos, C. J. Schinas, G. Stefanidou, On the recursive sequence $x_{n+1} = A+(x_{n-1}^p/x_n^q)$, Adv. Differ. Equ., 2009 (2009), 327649. https://doi.org/10.1155/2009/327649 doi: 10.1155/2009/327649
[23]	G. Papaschinopoulos, C. J. Schinas, G. Stefanidou, Two modifications of the Beverton–Holt equation, Int. J. Difference Equ., 4 (2009), 115–136.
[24]	G. Polya, G. Szegö, Problems and theorems in analysis I, Berlin: Springer, 1998. https://doi.org/10.1007/978-3-642-61983-0
[25]	V. A. Sadovnichiy, A. Grigoryan, S. Konyagin, Zadachi studencheskih matematichsikh olimpiad, (in Russian), Izdatel'stvo Moskovskogo Universiteta, 1987.
[26]	V. A. Sadovnichiy, A. S. Podkolzin, Zadachi studencheskih olimpiad po matematike, (in Russian), Moskva: Nauka, 1978.
[27]	S. Stević, On the recursive sequence $x_{n+1} = g(x_n, x_{n-1})/(A+x_n)$, Appl. Math. Lett., 15 (2002), 305–308. https://doi.org/10.1016/S0893-9659(01)00135-5 doi: 10.1016/S0893-9659(01)00135-5
[28]	S. Stević, Asymptotic behavior of a class of nonlinear difference equations, Discrete Dyn. Nat. Soc., 2006 (2006), 47156. https://doi.org/10.1155/DDNS/2006/47156 doi: 10.1155/DDNS/2006/47156
[29]	S. Stević, On monotone solutions of some classes of difference equations, Discrete Dyn. Nat. Soc., 2006 (2006), 53890. https://doi.org/10.1155/DDNS/2006/53890 doi: 10.1155/DDNS/2006/53890
[30]	S. Stević, Asymptotics of some classes of higher order difference equations, Discrete Dyn. Nat. Soc., 2007 (2007), 56813. https://doi.org/10.1155/2007/56813 doi: 10.1155/2007/56813
[31]	S. Stević, Nontrivial solutions of a higher-order rational difference equation, Math. Notes, 84 (2008), 718–724. https://doi.org/10.1134/S0001434608110138 doi: 10.1134/S0001434608110138
[32]	S. Stević, Complete picture on the monotonicity character of a class of sequences with a parameter, J. Math. Inequal., 19 (2025), 1093–1107. http://dx.doi.org/10.7153/jmi-2025-19-70 doi: 10.7153/jmi-2025-19-70
[33]	S. Stević, Convex combinations of some convergent sequences, Math. Methods Appl. Sci., 48 (2025), 2819–2832. https://doi.org/10.1002/mma.10463 doi: 10.1002/mma.10463
[34]	S. Stević, On some sequences whose limit is the Euler gamma constant, J. Inequal. Appl., 2025 (2025), 75. https://doi.org/10.1186/s13660-025-03321-7 doi: 10.1186/s13660-025-03321-7
[35]	S. Stević, B. Iričanin, W. Kosmala, Z. Šmarda, Monotonicity character of convex combinations of two sequences generated by the square of the cube root function, J. Inequal. Appl., 2025 (2025), 122. https://doi.org/10.1186/s13660-025-03375-7 doi: 10.1186/s13660-025-03375-7
[36]	R. P. Ushakov, Zadacha 3, In: Matematika segodnya '90, (in Russian), Kiev: Vishcha Shkola, 1990.
[37]	R. P. Ushakov, Reshenie Zadachi 3 iz Matematika Segodnya '90, In: Matematika segodnya '94, (in Russian), Kiev: Vishcha Shkola, 1994.
[38]	V. A. Zorich, Mathematical analysis I, 2 Eds., Berlin: Springer, 2015. https://doi.org/10.1007/978-3-662-48792-1

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