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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.18412 |
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| _version_ | 1866914578168807424 |
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| author | Piejko, K. Sokół, J. Trcabka-Wiȩc\a{l}aw, K. |
| author_facet | Piejko, K. Sokół, J. Trcabka-Wiȩc\a{l}aw, K. |
| contents | In this paper we consider some properties of Jackson's difference operator for convex univalent functions in $|z|<1$ with complex parameter $q$ as a Hadamard product of two power series. Jackson in 1908 introduced for a real $q$, $q\in[0,1)$, the difference operator \mbox{${\rm d}_qf(z)$} for an analytic function $f$ in the unit disc $|z|<1$ in the complex plane. Thanks to this operator, many mathematicians have extended the theory of functions in $q$-theory. The $q$-theory has found many applications in theory of hypergeometric series, special functions, combinatorics, number theory, fluid mechanics, quantum mechanics and physics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_18412 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Hadamard product of convex functions and Jackson operator Piejko, K. Sokół, J. Trcabka-Wiȩc\a{l}aw, K. Complex Variables 30C45 In this paper we consider some properties of Jackson's difference operator for convex univalent functions in $|z|<1$ with complex parameter $q$ as a Hadamard product of two power series. Jackson in 1908 introduced for a real $q$, $q\in[0,1)$, the difference operator \mbox{${\rm d}_qf(z)$} for an analytic function $f$ in the unit disc $|z|<1$ in the complex plane. Thanks to this operator, many mathematicians have extended the theory of functions in $q$-theory. The $q$-theory has found many applications in theory of hypergeometric series, special functions, combinatorics, number theory, fluid mechanics, quantum mechanics and physics. |
| title | Hadamard product of convex functions and Jackson operator |
| topic | Complex Variables 30C45 |
| url | https://arxiv.org/abs/2605.18412 |