Jordan-Type Inequalities and Stratification
Abstract
:1. Introduction
2. Preliminaries
3. Main Results
- (i) The family of functions
- (ii) The family of functions
- (ii) Since , we obtain the one-parameter family of functions:
- (i) If , then the lower bounds of the function are given by
- (ii) If , then the equality
- (iii) If , then the upper bounds of the function are given by
- (iv) Each function from the family , for , has exactly one maximum and exactly one minimum at certain points , respectively, on the interval . Additionally, it holds that . The function , for , has exactly one maximum on , and for has exactly one minimum on .
- (v) The equality
- (ii) It is easily seen that and . In part of this proof, it will be shown that each function , for , has exactly one maximum and exactly one minimum on the interval , respectively. Hence, the stated inequalities follow.
- (iii) The assertion is equivalent to for and . Continuing from part of this proof, using multiple applications of L’Hôpital’s rule, it can be shown that
- (iv) Let us examine the monotonicity of functions from the family for on . The fourth derivative of with respect to x is
- By analyzing the monotonicity of the functions , , , , and for and for , in a similar manner, it can be concluded that the function , for , has exactly one maximum on , while the function , for , has exactly one minimum on .
- Note that the infimum of the error , for , exists and is attained when
- (i) If , then the upper bounds of the function are given by
- (ii) If , then the equality
- (iii) If , then the lower bounds of the function are given by
- (iv) Each function from the family , for , has exactly one maximum at a point on the interval .
- (v) The equality
- (ii) Continuing from the previous part of the proof, , using multiple applications of L’Hôpital’s rule, it can be shown that
- (iii) The assertion is equivalent to for and . Let us notice that for , where . In Statement 1, it has already been proven that for on the interval . Given that the family of functions is increasingly stratified with respect to the parameter q based on Lemma 3, for , it will also hold that
- (iv) It has been established in part of the proof for . Similarly, the proof holds for .
- (v) Note that the infimum of the error , for , exists and is attained when
- (i) For and , it holds that
- (ii) For and , it holds that
4. Applications
4.1. Improvements of Theorems 2–5
- (i) If , then
- (ii) If , then the equality
- (iii) If , then
- (iv) Each function from the family , for , has exactly one maximum at a point on the interval .
- (v) The equality
- (ii) Let us examine the monotonicity of functions for on the interval in a similar manner as in the proof of Statement 1. The second derivative of with respect to x is
- (iii) The claim follows directly from Statement 2 and based on the stratification. Namely, for , it holds that .
- (iv) It has been proven within proof .
- (v) Note that the infimum of the error , for , exists and is attained when
- (i) If , then
- (ii) If , then the equality
- (iii) If , then
- (iv) Each function from the family , for , has exactly one minimum at a point on the interval .
- (v) The equality
- (ii) Let us examine the monotonicity of functions for on the interval in a similar manner as in the proof of Statement 1. The third derivative of with respect to x is
- (iii) The claim follows directly from Statement 1 and based on the stratification. Namely, for , it holds that .
- (iv) It has been proven within proof .
- (v) Note that the infimum of the error , for , exists and is attained when
- (i) If , then
- (ii) If , then the equality
- (iii) If , then
- (iv) Each function from the family , for , has exactly one minimum at a point on the interval .
- (v) The equality
- (i) If , then
- (ii) If , then the equality
- (iii) If , then
- (iv) Each function from the family , for , has exactly one minimum at a point on the interval .
- (v) The equality
4.2. Approximations of the Function
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Abbreviations
MTP | Mixed Trigonometric Polynomial |
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Upper Bound of the Function on the Interval | Maximum Deviation from the Function on the Interval |
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Lower Bound of the Function on the Interval | Maximum Deviation from the Function on the Interval |
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Minimax Approximation of the Function on the Interval | Maximum Deviation from the Function on the Interval |
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Mićović, M.; Malešević, B. Jordan-Type Inequalities and Stratification. Axioms 2024, 13, 262. https://doi.org/10.3390/axioms13040262
Mićović M, Malešević B. Jordan-Type Inequalities and Stratification. Axioms. 2024; 13(4):262. https://doi.org/10.3390/axioms13040262
Chicago/Turabian StyleMićović, Miloš, and Branko Malešević. 2024. "Jordan-Type Inequalities and Stratification" Axioms 13, no. 4: 262. https://doi.org/10.3390/axioms13040262