New Applications of the Bernardi Integral Operator
Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formu...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/8/7/1180 |
| _version_ | 1797562123320033280 |
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| author | Shigeyoshi Owa H. Özlem Güney |
| author_facet | Shigeyoshi Owa H. Özlem Güney |
| author_sort | Shigeyoshi Owa |
| collection | DOAJ |
| description | Let <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the class of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which are analytic <i>p</i>-valent functions in the closed unit disk <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi mathvariant="double-struck">U</mi> <mo>¯</mo> </mover> <mo>=</mo> <mfenced separators="" open="{" close="}"> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mfenced open="|" close="|"> <mi>z</mi> </mfenced> <mo>≤</mo> <mn>1</mn> </mfenced> </mrow> </semantics> </math> </inline-formula>. The expression <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mrow> <mo>−</mo> <mi>m</mi> <mo>−</mo> <mi>λ</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is defined by using fractional integrals of order <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> <mo>∈</mo> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> When <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> becomes Bernardi integral operator. Using the fractional integral <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mrow> <mo>−</mo> <mi>m</mi> <mo>−</mo> <mi>λ</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the subclass <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mfenced separators="" open="(" close=")"> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>ρ</mi> <mo>;</mo> <mi>m</mi> <mo>,</mo> <mi>λ</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is introduced. In the present paper, we discuss some interesting properties for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> concerning with the class <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mfenced separators="" open="(" close=")"> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>ρ</mi> <mo>;</mo> <mi>m</mi> <mo>,</mo> <mi>λ</mi> </mfenced> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Also, some interesting examples for our results will be considered. |
| first_indexed | 2024-03-10T18:24:01Z |
| format | Article |
| id | doaj.art-bd30784d923949fcb29dd72346cf21c8 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T18:24:01Z |
| publishDate | 2020-07-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-bd30784d923949fcb29dd72346cf21c82023-11-20T07:10:02ZengMDPI AGMathematics2227-73902020-07-0187118010.3390/math8071180New Applications of the Bernardi Integral OperatorShigeyoshi Owa0H. Özlem Güney1“1 Decembrie 1918” University Alba Iulia, 510009 Alba-Iulia, RomaniaDepartment of Mathematics, Faculty of Science Dicle University, 21280 Diyarbakır, TurkeyLet <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> be the class of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> which are analytic <i>p</i>-valent functions in the closed unit disk <inline-formula> <math display="inline"> <semantics> <mrow> <mover> <mi mathvariant="double-struck">U</mi> <mo>¯</mo> </mover> <mo>=</mo> <mfenced separators="" open="{" close="}"> <mi>z</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> <mo>:</mo> <mfenced open="|" close="|"> <mi>z</mi> </mfenced> <mo>≤</mo> <mn>1</mn> </mfenced> </mrow> </semantics> </math> </inline-formula>. The expression <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mrow> <mo>−</mo> <mi>m</mi> <mo>−</mo> <mi>λ</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> is defined by using fractional integrals of order <inline-formula> <math display="inline"> <semantics> <mi>λ</mi> </semantics> </math> </inline-formula> for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> <mo>∈</mo> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> When <inline-formula> <math display="inline"> <semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> and <inline-formula> <math display="inline"> <semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mrow> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula> becomes Bernardi integral operator. Using the fractional integral <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>B</mi> <mrow> <mo>−</mo> <mi>m</mi> <mo>−</mo> <mi>λ</mi> </mrow> </msub> <mi>f</mi> <mrow> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> <mo>,</mo> </mrow> </semantics> </math> </inline-formula> the subclass <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mfenced separators="" open="(" close=")"> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>ρ</mi> <mo>;</mo> <mi>m</mi> <mo>,</mo> <mi>λ</mi> </mfenced> </mrow> </semantics> </math> </inline-formula> of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>A</mi> <mo>(</mo> <mi>p</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> is introduced. In the present paper, we discuss some interesting properties for <inline-formula> <math display="inline"> <semantics> <mrow> <mi>f</mi> <mo>(</mo> <mi>z</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> concerning with the class <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>T</mi> <mrow> <mi>p</mi> <mo>,</mo> <mi>n</mi> </mrow> </msub> <mfenced separators="" open="(" close=")"> <msub> <mi>α</mi> <mi>s</mi> </msub> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>ρ</mi> <mo>;</mo> <mi>m</mi> <mo>,</mo> <mi>λ</mi> </mfenced> <mo>.</mo> </mrow> </semantics> </math> </inline-formula> Also, some interesting examples for our results will be considered.https://www.mdpi.com/2227-7390/8/7/1180analytic <i>p</i>-valent functionBernardi integral operatorLibera integral operatorfractional integralgamma functionMiller–Mocanu lemma |
| spellingShingle | Shigeyoshi Owa H. Özlem Güney New Applications of the Bernardi Integral Operator Mathematics analytic <i>p</i>-valent function Bernardi integral operator Libera integral operator fractional integral gamma function Miller–Mocanu lemma |
| title | New Applications of the Bernardi Integral Operator |
| title_full | New Applications of the Bernardi Integral Operator |
| title_fullStr | New Applications of the Bernardi Integral Operator |
| title_full_unstemmed | New Applications of the Bernardi Integral Operator |
| title_short | New Applications of the Bernardi Integral Operator |
| title_sort | new applications of the bernardi integral operator |
| topic | analytic <i>p</i>-valent function Bernardi integral operator Libera integral operator fractional integral gamma function Miller–Mocanu lemma |
| url | https://www.mdpi.com/2227-7390/8/7/1180 |
| work_keys_str_mv | AT shigeyoshiowa newapplicationsofthebernardiintegraloperator AT hozlemguney newapplicationsofthebernardiintegraloperator |