# Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus

^{1}

^{2}

^{3}

^{4}

^{5}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction and Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Lemma**

**1.**

## 2. Main Results

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Corollary**

**2.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**3.**

**Proof.**

**Corollary**

**4.**

**Theorem**

**3.**

**Proof.**

**Corollary**

**5.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Corollary**

**6.**

**Proof.**

## 3. Conclusions

- Sharper upper and lower bounds were obtained in terms of polynomials. New consequences of such sharper bounds are provided in the corollaries in terms of the integral estimate of ${\int}_{0}^{\frac{\pi}{2}}\frac{sin\left(x\right)}{x}dx$ and in terms of the fractional integral estimates of ${}_{a}{I}_{t}^{\alpha}\left(\frac{{e}^{x}+arctan\left(x\right)}{\sqrt{x}}\right)$ and ${}_{a}{I}_{t}^{\alpha}\left(\frac{cosh\left(x\right)}{x}\right)$.
- Question arises with respect to which would be the lowest upper and biggest lower bound for obtained inequalities, which leaves room for further research.
- Each of Theorem 2–4 can be easily generalized to arbitrary n as they rely on the remainder of Taylor expansion.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Mitrinović, D.S. Analytic Inequalities; Springer: Berlin/Heidelberg, Germany, 1970. [Google Scholar]
- Bullen, P.S. A Dictionary of Inequalities. In Pitman Monographs and Surveys in Pure and Applied Mathematics; Addison Wesley Longman Limited: Harlow, UK, 1998; Volume 97. [Google Scholar]
- Kober, H. Approximation by integral functions in the complex domain. Trans. Am. Math. Soc.
**1944**, 56, 7–31. [Google Scholar] [CrossRef] [Green Version] - Sándor, J. On the concavity of sin x/x. Octogon Math. Mag.
**2005**, 13, 406–407. [Google Scholar] - Bagul, Y.J.; Dhaigude, R.M.; Kostić, M.; Chesneau, C. Polynomial-Exponential Bounds for Some Trigonometric and Hyperbolic Functions. Axioms
**2021**, 10, 308. [Google Scholar] [CrossRef] - Chouikla, R.A.; Chesneau, C.; Yogesh, J.B. Some refinements of well-known inequalities involving trigonometric functions. J. Ramanujan Math. Soc.
**2021**, 36, 193–202. [Google Scholar] - Bagul, J.Y.; Chesneau, C. Generalized bounds for sine and cosine functions. Asian-Eur. J. Math.
**2022**, 15, 2250012. [Google Scholar] [CrossRef] - Dhaigude, M.R.; Yogesh, J.B. Simple efficient bounds for arcsine and arctangent functions. Punjab Univ. J. Math.
**2021**. [Google Scholar] [CrossRef] - Neuman, E. Refinements and generalizations of certain inequalities involving trigonometric and hyperbolic functions. Adv. Inequal. Appl.
**2012**, 1, 1–11. [Google Scholar] - Rodić, M. On the Converse Jensen-Type Inequality for Generalized f-Divergences and Zipf–Mandelbrot Law. Mathematics
**2022**, 10, 947. [Google Scholar] [CrossRef] - Rodić, M. Some Generalizations of the Jensen-Type Inequalities with Applications. Axioms
**2022**, 11, 227. [Google Scholar] [CrossRef] - Abramowitz, M.; Stegun, I.A. Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables; Dover Publications: New York, NY, USA, 1992. [Google Scholar]
- Hermann, R. Fractional Calculus An Introduction For Physicists; World Scientific Publishing Co. Pte. Ltd.: Singapore, 2011. [Google Scholar]
- Oldham, K.B.; Spanier, J. The Fractional Calculus Theory and Applications of Differentation and Integration to Arbitrary Order; Academic Press, Inc.: London, UK, 1974. [Google Scholar]
- Yang, X.J. General Fractional Derivatives Theory, Methods and Applications; Taylor and Francis Group: London, UK, 2019. [Google Scholar]
- Anderson, G.D.; Vamanamurthy, M.K.; Vuorinen, M. Conformal Invariants, Inequalities and Quasiconformal Maps; John Wiley and Sons: New York, NY, USA, 1997. [Google Scholar]

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Stojiljković, V.; Radojević, S.; Çetin, E.; Čavić, V.Š.; Radenović, S.
Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus. *Symmetry* **2022**, *14*, 1260.
https://doi.org/10.3390/sym14061260

**AMA Style**

Stojiljković V, Radojević S, Çetin E, Čavić VŠ, Radenović S.
Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus. *Symmetry*. 2022; 14(6):1260.
https://doi.org/10.3390/sym14061260

**Chicago/Turabian Style**

Stojiljković, Vuk, Slobodan Radojević, Eyüp Çetin, Vesna Šešum Čavić, and Stojan Radenović.
2022. "Sharp Bounds for Trigonometric and Hyperbolic Functions with Application to Fractional Calculus" *Symmetry* 14, no. 6: 1260.
https://doi.org/10.3390/sym14061260