Subtopic Deep Dive
Fractional Integrals Inequalities
Research Guide
What is Fractional Integrals Inequalities?
Fractional integrals inequalities establish upper and lower bounds for Riemann-Liouville and Hadamard fractional integrals under convexity, s-convexity, and harmonically convex assumptions.
This subtopic extends classical inequalities like Hermite-Hadamard, Ostrowski, and Hardy types to fractional integrals. Key works include Sarıkaya et al. (2012) with 890 citations on Hermite-Hadamard inequalities for fractional integrals. Sarıkaya and Yıldırım (2017) provide 239 citations on Riemann-Liouville variants.
Why It Matters
Fractional integrals inequalities bound operators in fractional calculus, essential for modeling anomalous diffusion in physics and viscoelasticity in engineering. Sarıkaya et al. (2012) inequalities apply to error estimates in numerical fractional differentiation. Kufner and Persson (2003) weighted Hardy inequalities support stability analysis in fractional differential equations used in heat transfer simulations.
Key Research Challenges
Extending to variable-order
Variable-order fractional integrals complicate convexity assumptions, requiring new bound techniques. Chen and Katugampola (2016) address generalized integrals but leave gaps in quantum variants. Open issues persist in uniform bounds across orders.
Quantum fractional variants
Quantum calculus introduces non-standard derivatives, challenging classical inequality proofs. Tariboon and Ntouyas (2014) extend Hermite-Hadamard to quantum integrals on finite intervals. Verification of sharpness remains unresolved for higher dimensions.
s-convexity in local fractals
Local fractional integrals on fractal sets demand adapted Ostrowski-type bounds. Sarıkaya and Budak (2016) establish generalized forms, citing 159 times. Proving optimality under s-convexity constraints is computationally intensive.
Essential Papers
Hermite–Hadamard’s inequalities for fractional integrals and related fractional inequalities
Mehmet Zeki Sarıkaya, Erhan Set, Hatice Yaldız et al. · 2012 · Mathematical and Computer Modelling · 890 citations
Weighted Inequalities of Hardy Type
Alois Kufner, Lars-Erik Persson · 2003 · WORLD SCIENTIFIC eBooks · 563 citations
Hardy's Inequality and Related Topics Some Weighted Norm Inequalities The Hardy-Steklov Operator Higher Order Hardy Inequalities Fractional Order Hardy Inequalities Integral Operators on the Cone o...
On Hermite-Hadamard type inequalities for Riemann-Liouville fractional integrals
Mehmet Zeki Sarıkaya, Hüseyin Yıldırım · 2017 · Miskolc mathematical notes/Mathematical notes · 239 citations
In this paper, we have established Hermite-Hadamard-type inequalities for fractional integrals and will be given an identity.With the help of this fractional-type integral identity, we give some in...
On Minkowski and Hermite-Hadamard integral inequalities via fractional integration
Zoubir Dahmani · 2010 · Annals of Functional Analysis · 226 citations
Hermite–Hadamard and Hermite–Hadamard–Fejér type inequalities for generalized fractional integrals
Hua Chen, Udita N. Katugampola · 2016 · Journal of Mathematical Analysis and Applications · 201 citations
Quantum integral inequalities on finite intervals
Jessada Tariboon, Sotiris K. Ntouyas · 2014 · Journal of Inequalities and Applications · 200 citations
In this paper, some of the most important integral inequalities of analysis are extended to quantum calculus. These include the Hölder, Hermite-Hadamard, trapezoid, Ostrowski, Cauchy-Bunyakovsky-Sc...
New inequalities of Ostrowski type for mappings whose derivatives are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mi>s</mml:mi></mml:math>-convex in the second sense via fractional integrals
Erhan Set · 2011 · Computers & Mathematics with Applications · 198 citations
Reading Guide
Foundational Papers
Start with Sarıkaya et al. (2012, 890 citations) for core Hermite-Hadamard fractional inequalities; follow Kufner and Persson (2003, 563 citations) for weighted Hardy context; Dahmani (2010, 226 citations) for Minkowski via fractional integration.
Recent Advances
Sarıkaya and Yıldırım (2017, 239 citations) on Riemann-Liouville refinements; Chen and Katugampola (2016, 201 citations) on generalized Fejér types; Sarıkaya and Budak (2016, 159 citations) on local fractal Ostrowski.
Core Methods
Riemann-Liouville and Hadamard operators under convexity; quantum calculus integrals (Tariboon and Ntouyas, 2014); s-convexity for Ostrowski (Set, 2011); weighted norms for Hardy (Kufner and Persson, 2003).
How PapersFlow Helps You Research Fractional Integrals Inequalities
Discover & Search
Research Agent uses citationGraph on Sarıkaya et al. (2012, 890 citations) to map extensions like Set (2011) and Işcan and Wu (2014). exaSearch queries 'Hermite-Hadamard fractional Riemann-Liouville convexity' retrieves 200+ papers; findSimilarPapers links Dahmani (2010) to quantum variants by Tariboon and Ntouyas (2014).
Analyze & Verify
Analysis Agent applies readPaperContent to extract convexity conditions from Sarıkaya and Yıldırım (2017), then verifyResponse with CoVe checks inequality sharpness against Kufner and Persson (2003) weights. runPythonAnalysis numerically verifies Ostrowski bounds from Set (2011) using NumPy for s-convex test functions; GRADE scores evidence strength on fractional order convergence.
Synthesize & Write
Synthesis Agent detects gaps in quantum extensions beyond Tariboon and Ntouyas (2014), flagging contradictions in harmonically convex proofs from Işcan and Wu (2014). Writing Agent uses latexEditText to draft proofs, latexSyncCitations for 10+ references, and latexCompile for publication-ready manuscripts; exportMermaid visualizes inequality hierarchies.
Use Cases
"Numerically verify Hermite-Hadamard inequality for Riemann-Liouville integral on s-convex functions."
Research Agent → searchPapers 'Sarıkaya fractional Hermite-Hadamard' → Analysis Agent → readPaperContent (Sarıkaya et al. 2012) → runPythonAnalysis (NumPy plot of bound errors for alpha=0.5) → matplotlib graph of convergence.
"Write LaTeX proof extending Ostrowski inequality to Hadamard fractional integrals."
Synthesis Agent → gap detection on Sarıkaya and Budak (2016) → Writing Agent → latexEditText (insert convexity lemma) → latexSyncCitations (add Set 2011) → latexCompile → PDF with compiled fractional bounds.
"Find GitHub code for simulating quantum fractional inequalities."
Research Agent → searchPapers 'Tariboon quantum integral inequalities' → Code Discovery → paperExtractUrls (Tariboon and Ntouyas 2014) → paperFindGithubRepo → githubRepoInspect → Python scripts for Hölder quantum bounds.
Automated Workflows
Deep Research workflow scans 50+ papers via citationGraph from Sarıkaya et al. (2012), producing structured report on convexity types with GRADE scores. DeepScan's 7-step chain verifies Sarıkaya and Yıldırım (2017) identities: readPaperContent → runPythonAnalysis → CoVe checkpoints. Theorizer generates conjectures for variable-order gaps from Chen and Katugampola (2016).
Frequently Asked Questions
What defines fractional integrals inequalities?
Bounds on Riemann-Liouville or Hadamard fractional integrals for convex, s-convex, or harmonically convex functions, extending Hermite-Hadamard and Ostrowski types (Sarıkaya et al., 2012).
What are main methods used?
Fractional integral identities with convexity assumptions prove bounds; examples include quantum extensions (Tariboon and Ntouyas, 2014) and weighted Hardy operators (Kufner and Persson, 2003).
What are key papers?
Sarıkaya et al. (2012, 890 citations) on Hermite-Hadamard; Kufner and Persson (2003, 563 citations) on fractional Hardy; Sarıkaya and Yıldırım (2017, 239 citations) on Riemann-Liouville.
What open problems exist?
Sharpness in variable-order and quantum settings; optimal constants for local fractal integrals (Sarıkaya and Budak, 2016); multidimensional extensions.
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