A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices
In this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the growth in the Dirichlet series with s...
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MDPI AG
2022-06-01
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| author | Hongyan Xu Guangsheng Chen Hari Mohan Srivastava Hong Li Zuxing Xuan Yongqin Cui |
| author_facet | Hongyan Xu Guangsheng Chen Hari Mohan Srivastava Hong Li Zuxing Xuan Yongqin Cui |
| author_sort | Hongyan Xu |
| collection | DOAJ |
| description | In this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the growth in the Dirichlet series with smaller indices if these Dirichlet series have different growth indices; (ii) the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mn>1</mn></msub></semantics></math></inline-formula>-type of the Hadamard–Kong product series is equal to zero if <i>p</i> Dirichlet series are of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mi>j</mi></msub></semantics></math></inline-formula>-regular growth, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>1</mn></msub><mo><</mo><msub><mi>q</mi><mn>2</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mi>q</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mi>j</mi></msub><mo>∈</mo><msub><mi>N</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi></mrow></semantics></math></inline-formula>. The second purpose is to reveal the properties of the growth in the Hadamard–Kong product series of two Dirichlet series—when one Dirichlet series is of finite order, the other is of logarithmic order, and two Dirichlet series are of finite logarithmic order—and obtain the growth relationships between the Hadamard–Kong product series and two Dirchlet series concerning the order, the logarithmic order, logarithmic type, etc. Finally, some examples are given to show that our results are best possible. |
| first_indexed | 2024-03-09T10:27:16Z |
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| spelling | doaj.art-756aea18a7b544488b33eb59f00a6e932023-12-01T21:35:09ZengMDPI AGMathematics2227-73902022-06-011013222010.3390/math10132220A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth IndicesHongyan Xu0Guangsheng Chen1Hari Mohan Srivastava2Hong Li3Zuxing Xuan4Yongqin Cui5College of Arts and Sciences, Suqian University, Suqian 223800, ChinaCollege of Mathematics and Computer Science, Guangxi Science & Technology Normal University, Laibin 546199, ChinaDepartment of Mathematics and Statistics, University of Victoria, Victoria, BC V8W 3R4, CanadaSchool of Mathematics and Computer Science, Gannan Normal University, Ganzhou 341000, ChinaDepartment of General Education, Beijing Union University, Beijing 100101, ChinaDepartment of Informatics and Engineering, Jingdezhen Ceramic University, Jingdezhen 333403, ChinaIn this paper, our first purpose is to describe a class of phenomena involving the growth in the Hadamard–Kong product of several Dirichlet series with different growth indices. We prove that (i) the order of the Hadamard–Kong product series is determined by the growth in the Dirichlet series with smaller indices if these Dirichlet series have different growth indices; (ii) the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mn>1</mn></msub></semantics></math></inline-formula>-type of the Hadamard–Kong product series is equal to zero if <i>p</i> Dirichlet series are of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>q</mi><mi>j</mi></msub></semantics></math></inline-formula>-regular growth, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>1</mn></msub><mo><</mo><msub><mi>q</mi><mn>2</mn></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mi>q</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mi>j</mi></msub><mo>∈</mo><msub><mi>N</mi><mo>+</mo></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>j</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>p</mi></mrow></semantics></math></inline-formula>. The second purpose is to reveal the properties of the growth in the Hadamard–Kong product series of two Dirichlet series—when one Dirichlet series is of finite order, the other is of logarithmic order, and two Dirichlet series are of finite logarithmic order—and obtain the growth relationships between the Hadamard–Kong product series and two Dirchlet series concerning the order, the logarithmic order, logarithmic type, etc. Finally, some examples are given to show that our results are best possible.https://www.mdpi.com/2227-7390/10/13/2220Dirichlet seriesHadamard–Kong productgrowth |
| spellingShingle | Hongyan Xu Guangsheng Chen Hari Mohan Srivastava Hong Li Zuxing Xuan Yongqin Cui A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices Mathematics Dirichlet series Hadamard–Kong product growth |
| title | A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices |
| title_full | A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices |
| title_fullStr | A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices |
| title_full_unstemmed | A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices |
| title_short | A Study of the Growth Results for the Hadamard Product of Several Dirichlet Series with Different Growth Indices |
| title_sort | study of the growth results for the hadamard product of several dirichlet series with different growth indices |
| topic | Dirichlet series Hadamard–Kong product growth |
| url | https://www.mdpi.com/2227-7390/10/13/2220 |
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