Fejér-Type Inequalities for Some Classes of Differentiable Functions
We let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> be a convex function on an interval <inline-formula><math xmlns="http:/...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-09-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/17/3764 |
| _version_ | 1797582173058891776 |
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| author | Bessem Samet |
| author_facet | Bessem Samet |
| author_sort | Bessem Samet |
| collection | DOAJ |
| description | We let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> be a convex function on an interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><msub><mi>ι</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ι</mi><mn>2</mn></msub><mo>]</mo><mo>⊂</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mrow><mo>[</mo><msub><mi>ι</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ι</mi><mn>2</mn></msub><mo>]</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> is symmetric with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><msub><mi>ι</mi><mn>1</mn></msub><mo>+</mo><msub><mi>ι</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>υ</mi><mfenced separators="" open="(" close=")"><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></msubsup><msub><mi>ι</mi><mi>j</mi></msub></mfenced><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>1</mn></msub></mrow><msub><mi>ι</mi><mn>2</mn></msub></msubsup><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>s</mi><mo>≤</mo></mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>1</mn></msub></mrow><msub><mi>ι</mi><mn>2</mn></msub></msubsup><mi>υ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>s</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mi>υ</mi><mrow><mo>(</mo><msub><mi>ι</mi><mi>j</mi></msub><mo>)</mo></mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>1</mn></msub></mrow><msub><mi>ι</mi><mn>2</mn></msub></msubsup><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>s</mi><mo>.</mo></mrow></semantics></math></inline-formula> The above estimates were obtained by Fejér in 1906 as a generalization of the Hermite–Hadamard inequality (the above inequality with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>≡</mo><mn>1</mn></mrow></semantics></math></inline-formula>). This work is focused on the study of right-side Fejér-type inequalities in one- and two-dimensional cases for new classes of differentiable functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula>. In the one-dimensional case, the obtained results hold without any symmetry condition imposed on the weight function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>. In the two-dimensional case, the right side of Fejer’s inequality is extended to the class of subharmonic functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> on a disk. |
| first_indexed | 2024-03-10T23:17:09Z |
| format | Article |
| id | doaj.art-2f848cd130cd484dafd9d3d6c0c028cb |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T23:17:09Z |
| publishDate | 2023-09-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-2f848cd130cd484dafd9d3d6c0c028cb2023-11-19T08:31:48ZengMDPI AGMathematics2227-73902023-09-011117376410.3390/math11173764Fejér-Type Inequalities for Some Classes of Differentiable FunctionsBessem Samet0Department of Mathematics, College of Science, King Saud University, Riyadh 11451, Saudi Arabia We let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> be a convex function on an interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>[</mo><msub><mi>ι</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ι</mi><mn>2</mn></msub><mo>]</mo><mo>⊂</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>. If <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>∈</mo><mi>C</mi><mo>(</mo><mrow><mo>[</mo><msub><mi>ι</mi><mn>1</mn></msub><mo>,</mo><msub><mi>ι</mi><mn>2</mn></msub><mo>]</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>≥</mo><mn>0</mn></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula> is symmetric with respect to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mfrac><mrow><msub><mi>ι</mi><mn>1</mn></msub><mo>+</mo><msub><mi>ι</mi><mn>2</mn></msub></mrow><mn>2</mn></mfrac></semantics></math></inline-formula>, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mi>υ</mi><mfenced separators="" open="(" close=")"><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></msubsup><msub><mi>ι</mi><mi>j</mi></msub></mfenced><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>1</mn></msub></mrow><msub><mi>ι</mi><mn>2</mn></msub></msubsup><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>s</mi><mo>≤</mo></mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>1</mn></msub></mrow><msub><mi>ι</mi><mn>2</mn></msub></msubsup><mi>υ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>s</mi><mo>≤</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mn>2</mn></msubsup><mi>υ</mi><mrow><mo>(</mo><msub><mi>ι</mi><mi>j</mi></msub><mo>)</mo></mrow><msubsup><mo>∫</mo><mrow><msub><mi>ι</mi><mn>1</mn></msub></mrow><msub><mi>ι</mi><mn>2</mn></msub></msubsup><mi>ζ</mi><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mspace width="0.166667em"></mspace><mi>d</mi><mi>s</mi><mo>.</mo></mrow></semantics></math></inline-formula> The above estimates were obtained by Fejér in 1906 as a generalization of the Hermite–Hadamard inequality (the above inequality with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>ζ</mi><mo>≡</mo><mn>1</mn></mrow></semantics></math></inline-formula>). This work is focused on the study of right-side Fejér-type inequalities in one- and two-dimensional cases for new classes of differentiable functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula>. In the one-dimensional case, the obtained results hold without any symmetry condition imposed on the weight function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ζ</mi></semantics></math></inline-formula>. In the two-dimensional case, the right side of Fejer’s inequality is extended to the class of subharmonic functions <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>υ</mi></semantics></math></inline-formula> on a disk.https://www.mdpi.com/2227-7390/11/17/3764Fejér inequalityHermite–Hadamard inequalityconvex functionsdifferentiable functionssubharmonic functions |
| spellingShingle | Bessem Samet Fejér-Type Inequalities for Some Classes of Differentiable Functions Mathematics Fejér inequality Hermite–Hadamard inequality convex functions differentiable functions subharmonic functions |
| title | Fejér-Type Inequalities for Some Classes of Differentiable Functions |
| title_full | Fejér-Type Inequalities for Some Classes of Differentiable Functions |
| title_fullStr | Fejér-Type Inequalities for Some Classes of Differentiable Functions |
| title_full_unstemmed | Fejér-Type Inequalities for Some Classes of Differentiable Functions |
| title_short | Fejér-Type Inequalities for Some Classes of Differentiable Functions |
| title_sort | fejer type inequalities for some classes of differentiable functions |
| topic | Fejér inequality Hermite–Hadamard inequality convex functions differentiable functions subharmonic functions |
| url | https://www.mdpi.com/2227-7390/11/17/3764 |
| work_keys_str_mv | AT bessemsamet fejertypeinequalitiesforsomeclassesofdifferentiablefunctions |