Frame’s Types of Inequalities and Stratification

Branko Malešević; Dimitrije Jovanović

doi:10.56754/0719-0646.2601.001

DOI:

https://doi.org/10.56754/0719-0646.2601.001

Abstract

In this paper we examine some inequalities of Frame's type on the interval \((0,\pi/2)\). By observing this domain we simply obtain the results using the appropriate families of stratified functions and MTP - Mixed Trigonometric Polynomials. Additionally, from those families we specify a minimax approximant as a function with some optimal properties.

Keywords

Frame’s type inequalities , stratified families of functions , mixed trigonometric polynomial functions

Mathematics Subject Classification:

33B10 , 26D05

Branko Malešević School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia. https://orcid.org/0000-0002-4963-4149
Dimitrije Jovanović School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11000 Belgrade, Serbia. https://orcid.org/0009-0001-5842-9139

Pages: 1–19
Date Published: 2024-03-19
Vol. 26 No. 1 (2024)

“Queries—replies,” Mathematical Tables and other Aids to Computation, vol. 3, no. 28, pp. 561–563, 1949.

M. J. Cloud, B. C. Drachman, and L. P. Lebedev, Inequalities. With applications to engineering, 2nd ed. Springer, Cham, 2014, doi: 10.1007/978-3-319-05311-0.

N. Cutland, Computability. An introduction to recursive function theory. Cambridge University Press, Cambridge-New York, 1980.

B. Dong, B. Yu, and Y. Yu, “A symmetric homotopy and hybrid polynomial system solving method for mixed trigonometric polynomial systems,” Math. Comp., vol. 83, no. 288, pp. 1847–1868, 2014, doi: 10.1090/S0025-5718-2013-02763-9.

G. T. Freitas De Abreu, “Jensen-Cotes upper and lower bounds on the Gaussian Q-function and related functions”, IEEE Transactions on Communications, vol. 57, no. 11, pp. 3328–3338, 2009, doi: 10.1109/TCOMM.2009.11.080479.

Y. Lv, G. Wang, and Y. Chu, “A note on Jordan type inequalities for hyperbolic functions,” Appl. Math. Lett., vol. 25, no. 3, pp. 505–508, 2012, doi: 10.1016/j.aml.2011.09.046.

B. Malešević, T. Lutovac, and B. Banjac, “One method for proving some classes of exponential analytical inequalities,” Filomat, vol. 32, no. 20, pp. 6921–6925, 2018, doi: 10.2298/fil1820921m.

B. Malešević and M. Makragić, “A method for proving some inequalities on mixed trigonometric polynomial functions,” J. Math. Inequal., vol. 10, no. 3, pp. 849–876, 2016, doi: 10.7153/jmi-10-69.

B. Malešević and B. Mihailović, “A minimax approximant in the theory of analytic inequalities,” Appl. Anal. Discrete Math., vol. 15, no. 2, pp. 486–509, 2021, doi: 10.2298/aadm210511032m.

B. Malešević, M. Nenezić, L. Zhu, B. Banjac and M. Petrović, “Some new estimates of precision of Cusa-Huygens and Huygens approximations,” Appl. Anal. Discrete Math., vol. 15, no. 1, pp. 243–259, 2021, doi: 10.2298/aadm190904055m.

D. S. Mitrinović, Analytic inequalities, ser. Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Berlin, 1970, vol. 165.

G. Rahmatollahi and G. Abreu, “Closed-form hop-count distributions in random networks with arbitrary routing,” IEEE Transactions on Communications, vol. 60, no. 2, pp. 429–444, 2012, doi: 10.1109/TCOMM.2012.010512.110125.

T. Sch., “Literaturberichte: Konstruktionen und Approximationen,” Monatsh. Math. Phys., vol. 23, no. 1, p. A61, 1912, doi: 10.1007/BF01707814.

P. C. Sturm, Mémoire sur la résolution des équations numériques. Basel: Birkhäuser Basel, 2009, pp. 345–390, doi: 10.1007/978-3-7643-7990-2_29.

Z.-H. Yang, Y.-M. Chu, Y.-Q. Song and Y.-M. Li, “A sharp double inequality for trigonometric functions and its applications,” Abstr. Appl. Anal., 2014, Art. ID 592085, doi: 10.1155/2014/592085.

Z.-H. Yang, Y.-M. Chu, and X.-H. Zhang, “Sharp Cusa type inequalities with two parameters and their applications,” Appl. Math. Comput., vol. 268, pp. 1177–1198, 2015, doi: 10.1016/j.amc.2015.07.025.

Z.-H. Yang, Y.-L. Jiang, Y.-Q. Song, and Y.-M. Chu, “Sharp inequalities for trigonometric functions,” Abstr. Appl. Anal., 2014, Art. ID 601839, doi: 10.1155/2014/601839.

T.-H. Zhao and Y.-M. Chu, “Some general Wilker-Huygens inequalities,” Appl. Anal. Discrete Math., vol. 16, no. 2, pp. 400–426, 2022.

L. Zhu, “On Frame’s inequalities,” J. Inequal. Appl., 2018, Art. ID 94, doi: 10.1186/s13660- 018-1687-x.

L. Zhu and B. Malešević, “New inequalities of Huygens-type involving tangent and sine functions,” Hacet. J. Math. Stat., vol. 52, no. 1, pp. 36–61, 2023, doi: 10.1007/s00182-022-00809-0.