Abstract
We present uniform and $L_p$ mixed Caputo-Bochner abstract generalized fractional Landau inequalities over $\mathbb{R}$ of fractional orders $2 < \alpha \leq 3 $. These estimate the size of first and second derivatives of a composition with a Banach space valued function over $\mathbb{R}$. We give applications when $α = 2.5$.
Keywords
Abstract generalized fractional Landau inequality right and left Caputo abstract generalized fractional derivatives fractional derrivatives
References
- A. Aglic Aljinovic, Lj. Marangunic and J. Pecaric: On Landau type inequalities via Ostrowski inequalities, Nonlinear Funct. Anal. Appl. Vol., 10, 4 (2005), pp. 565-579.
- G. Anastassiou: Fractional Differentiation inequalities, Research monograph, Springer, New York, 2009.
- G. A. Anastassiou: Advances on Fractional Inequalities, Springer, New York, 2011.
- G. A. Anastassiou: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, 2018.
- N. S. Barnett, S. S. Dragomir: Some Landau type inequalities for functions whose derivatives are of locally bounded variation, Tamkang Journal of Mathematics, 37, No. 4, 301-308, winter 2006.
- Z. Ditzian: Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities, Math. Inequal. Appl., 3 (2000), 15-24.
- G. H. Hardy, J. E. Littlewood: Some integral inequalities connected with the calculus of variations, Quart. J. Math. Oxford Ser., 3 (1932), 241-252.
- G. H. Hardy, E. Landau and J. E. Littlewood: Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math. Z., 39 (1935), 677-695.
- R. R. Kallman, G. C. Rota: On the inequality $\Vert f' \Vert^2 \leq 4\Vert f'' \Vert$, in Inequalities, Vol. II, (O. Shisha, Ed.), 187-192, Academic Press, New York, 1970.
- E. Landau: Einige Ungleichungen für zweimal differentzierban funktionen, Proc. London Math. Soc., 13 (1913), 43-49.
- J. Mikusinski: The Bochner integral, Academic Press, New York, 1978.
- G. E. Shilov: Elementary Functional Analysis, Dover Publications Inc., New York, 19
Abstract
References
- A. Aglic Aljinovic, Lj. Marangunic and J. Pecaric: On Landau type inequalities via Ostrowski inequalities, Nonlinear Funct. Anal. Appl. Vol., 10, 4 (2005), pp. 565-579.
- G. Anastassiou: Fractional Differentiation inequalities, Research monograph, Springer, New York, 2009.
- G. A. Anastassiou: Advances on Fractional Inequalities, Springer, New York, 2011.
- G. A. Anastassiou: Intelligent Computations: Abstract Fractional Calculus, Inequalities, Approximations, Springer, Heidelberg, New York, 2018.
- N. S. Barnett, S. S. Dragomir: Some Landau type inequalities for functions whose derivatives are of locally bounded variation, Tamkang Journal of Mathematics, 37, No. 4, 301-308, winter 2006.
- Z. Ditzian: Remarks, questions and conjectures on Landau-Kolmogorov-type inequalities, Math. Inequal. Appl., 3 (2000), 15-24.
- G. H. Hardy, J. E. Littlewood: Some integral inequalities connected with the calculus of variations, Quart. J. Math. Oxford Ser., 3 (1932), 241-252.
- G. H. Hardy, E. Landau and J. E. Littlewood: Some inequalities satisfied by the integrals or derivatives of real or analytic functions, Math. Z., 39 (1935), 677-695.
- R. R. Kallman, G. C. Rota: On the inequality $\Vert f' \Vert^2 \leq 4\Vert f'' \Vert$, in Inequalities, Vol. II, (O. Shisha, Ed.), 187-192, Academic Press, New York, 1970.
- E. Landau: Einige Ungleichungen für zweimal differentzierban funktionen, Proc. London Math. Soc., 13 (1913), 43-49.
- J. Mikusinski: The Bochner integral, Academic Press, New York, 1978.
- G. E. Shilov: Elementary Functional Analysis, Dover Publications Inc., New York, 19
Details
| Primary Language | English |
|---|---|
| Subjects | Mathematical Sciences |
| Journal Section | Research Article |
| Authors | |
| Publication Date | March 1, 2021 |
| DOI | https://doi.org/10.33205/cma.764161 |
| IZ | https://izlik.org/JA38NG72GS |
| Published in Issue | Year 2021 Volume: 4 Issue: 1 |
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