Some Properties of a Falling Function and Related Inequalities on Green’s Functions
Abstract
:1. Introduction
2. Delta Fractional Operators and the Basic Lemmas
2.1. Delta Fractional Operators
- For , we have
- For as , we have
- t , we have
2.2. Basic Lemmas
- (a)
- The identity ξ in Theorem 2 can be expressed as follows:
- (b)
- The identities and in Theorem 3 can be expressed as follows:
- (i)
- If , then
- The function is decreasing with respect to r, for .
- The function is increasing with respect to t, for .
- (ii)
- If , then
- , for .
- , for .
- (iii)
- If , then
- The function is increasing with respect to r, for .
- The function is increasing with respect to t, for .
- (iv)
- If , then the function is non-decreasing with respect to t, for .
3. Taylor Falling Function and Green’s Function Results
3.1. Taylor Falling Function
- 1.
- .
- 2.
- , where , and , for , or, specifically, .
- 3.
- The function is nonincreasing with respect to t, for .
- 4.
- The function is nonincreasing with respect to t, for .
3.2. Green’s Function Results
- 1.
- , and then .
- 2.
- such that .
- 3.
- such that .
- For : According to Lemma 4, . Therefore, .
- For : By using Lemma 4, such that . Hence, .
- For : Again, by using Lemma 4, and such that . These lead to .
- For : We considerby Lemma 4, and by Theorem 4. Therefore, we get as desired.
- 1.
- is an increasing function with respect to t such that .
- 2.
- is an increasing function with respect to t such that .
- For : Clearly, from Lemma (4), such that implies that .
- For : We considerBy considering Lemma (4), we see that . Moreover, by considering Theorem 4, we have . Therefore, .
- For : According to Lemma (4), it can be seen that and such that implies that .
4. Application
5. Concluding Remarks
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Mohammed, P.O.; Agarwal, R.P.; Yousif, M.A.; Al-Sarairah, E.; Mahmood, S.A.; Chorfi, N. Some Properties of a Falling Function and Related Inequalities on Green’s Functions. Symmetry 2024, 16, 337. https://doi.org/10.3390/sym16030337
Mohammed PO, Agarwal RP, Yousif MA, Al-Sarairah E, Mahmood SA, Chorfi N. Some Properties of a Falling Function and Related Inequalities on Green’s Functions. Symmetry. 2024; 16(3):337. https://doi.org/10.3390/sym16030337
Chicago/Turabian StyleMohammed, Pshtiwan Othman, Ravi P. Agarwal, Majeed A. Yousif, Eman Al-Sarairah, Sarkhel Akbar Mahmood, and Nejmeddine Chorfi. 2024. "Some Properties of a Falling Function and Related Inequalities on Green’s Functions" Symmetry 16, no. 3: 337. https://doi.org/10.3390/sym16030337