Weierstrass functions and a generalization of the additive-multiplicative Weierstrass inequality
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912748981452800 |
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| author | Wiśniewska, Halina |
| author_facet | Wiśniewska, Halina |
| contents | Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq f(x)f(y). $$ By means of such functions we can extend the classical Weierstrass inequality (the above inequality for $f(x) = x$) to some trigonometric, Euler gamma, and log functions. Utilizing the Weierstrass property of $f(x) = \frac{\ln (1+x)}{\ln2}$, we obtain a new multiplicative inequality which, in turn, generalizes the classical Weierstrass inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_04650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weierstrass functions and a generalization of the additive-multiplicative Weierstrass inequality Wiśniewska, Halina Classical Analysis and ODEs 26A09, 26D05, 33B10, 33B15 Let $J$ denote the interval either $(0,1]$ or $ [1, \infty)$. A positive function $f$ on $J$ with $f(1) =1$ is reffered to as a Weierstrass function if it fulfils the double inequality for $x,y \in J$: $$f(x) + f(y) -1 \leq f(xy) \leq f(x)f(y). $$ By means of such functions we can extend the classical Weierstrass inequality (the above inequality for $f(x) = x$) to some trigonometric, Euler gamma, and log functions. Utilizing the Weierstrass property of $f(x) = \frac{\ln (1+x)}{\ln2}$, we obtain a new multiplicative inequality which, in turn, generalizes the classical Weierstrass inequality. |
| title | Weierstrass functions and a generalization of the additive-multiplicative Weierstrass inequality |
| topic | Classical Analysis and ODEs 26A09, 26D05, 33B10, 33B15 |
| url | https://arxiv.org/abs/2512.04650 |