Logarithmically Absolutely Monotone Trigonometric Functions

We study absolute monotonicity and logarithmic absolute monotonicity for the functions f z = cos α 1 z ⋅ ⋅ ⋯ cos α M z ⋅ sin β 1 z ⋅ ⋅ ⋯ sin β N z cos α ′ 1 z ⋅ ⋅ ⋯ cos α ′ M ′ z ⋅ sin β ′ 1 z ⋅ ⋅ ⋯ sin β ′ N ′ z zN ′ − N , where N,M,N ′ ,M ′ ∈ ℤ + , α j , α′ j , β j , β′ j ≥ 0. If β = 0, then the f...

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Bibliographic Details
Published in:Journal of mathematical sciences (New York, N.Y.) N.Y.), 2022-12, Vol.268 (6), p.773-782
Main Author: Vinogradov, O. L.
Format: Article
Language:English
Subjects:
Entire functions
Logarithms
Mathematical analysis
Mathematics
Mathematics and Statistics
Polynomials
Trigonometric functions
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Summary:We study absolute monotonicity and logarithmic absolute monotonicity for the functions f z = cos α 1 z ⋅ ⋅ ⋯ cos α M z ⋅ sin β 1 z ⋅ ⋅ ⋯ sin β N z cos α ′ 1 z ⋅ ⋅ ⋯ cos α ′ M ′ z ⋅ sin β ′ 1 z ⋅ ⋅ ⋯ sin β ′ N ′ z zN ′ − N , where N,M,N ′ ,M ′ ∈ ℤ + , α j , α′ j , β j , β′ j ≥ 0. If β = 0, then the factor sin βz is replaced by z . If N , M , N ′, or M ′ equals zero, then the corresponding factors are absent. A criterion of logarithmic absolute monotonicity for f is obtained. We give certain applications of absolute monotonicity to sharp inequalities for derivatives and differences of trigonometric polynomials and entire functions of exponential type.
ISSN:1072-3374
1573-8795
DOI:10.1007/s10958-022-06217-9