By Panos M. Pardalos, Themistocles M. Rassias
The contributions during this quantity objective to deepen knowing of a few of the present examine difficulties and theories in sleek subject matters reminiscent of calculus of adaptations, optimization concept, complicated research, genuine research, differential equations, and geometry. functions to those parts of arithmetic are awarded in the vast spectrum of analysis in Engineering technology with specific emphasis on equilibrium difficulties, complexity in numerical optimization, dynamical platforms, non-smooth optimization, advanced community research, statistical types and information mining, and effort platforms. extra emphasis is given to interdisciplinary examine, even supposing topics are taken care of in a unified and self-contained demeanour. The presentation of equipment, concept and purposes makes this tribute a useful reference for lecturers, researchers, and different execs attracted to natural and utilized learn, philosophy of arithmetic, and arithmetic schooling. a few overview papers released during this quantity can be really precious for a broader viewers of readers in addition to for graduate scholars who look for the most recent info.
Constantin Carathéodory’s wide-ranging impact within the foreign mathematical neighborhood was once noticeable in the course of the first Fields Medals awards on the foreign Congress of Mathematicians, Oslo, 1936. medals have been offered, one to Lars V. Ahlfors and one to Jesse Douglass. It used to be Carathéodory who awarded either their works throughout the beginning of the foreign Congress. This quantity comprises major papers in technological know-how and Engineering devoted to the reminiscence of Constantin Carathéodory and the spirit of his mathematical impact.
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Extra info for Contributions in Mathematics and Engineering: In Honor of Constantin Carathéodory
Math. Anal. Appl. 382, 814–821 (2011) 15. : On two different concept of superquadracity. Inequalities and Applications, pp. 209–215. Birkhauser, Basel (2010) 16. : Notes on a theorem of Hilbert. Math. Z. 6, 314–317 (1920) 17. : On the Jensen-Steffensen inequality and superquadracity. Anale Universitatii, Oradea, Fasc. Mathematica, Tom XVIII, pp. 269–275 (2011) 18. : Refinement of Hardy’s inequalities via superquadratic and subquadratic functions. J. Math. Anal. Appl. 339, 1305–1312 (2008) 19. : Hardy type inequalities via convexity-the journey so far.
Theorem 1 (Variation of Constants). 0; 1 and s; t 2 Œ0; 1/. t; /f . t; / is the Cauchy function for (3). 1 Taylor’s Formula and Integral Inequalities for Conformable Fractional Derivatives 27 Proof. t; /f . t; /f . t; /f . t; /f . t; /f . t/; s t u and the proof is complete. Theorem 2 (Taylor Formula). 0; 1 and n 2 N. n C 1/ times ˛-fractional differentiable on Œ0; 1/, and s; t 2 Œ0; 1/. s/ C kŠ ˛ nŠ s ˛ kD0 ˛ Ãn DnC1 ˛ f . /d˛ : Proof. t/. t/ C 1 nŠ Z tÂ s ˛ Ãn t˛ ˛ g. R. s/ ˛ m/Š m for 0 Ä m Ä n.
We consequently have that w also solves (5), and thus u Á w by uniqueness. u t Corollary 1. 0; 1 and s; r 2 Œ0; 1/ be fixed. n k/Š t˛ s˛ Ãk Â ˛ s˛ r˛ Ãn ˛ Proof. t/ D Taylor’s formula. It can also be shown directly. k : 1 nŠ t˛ r˛ n ˛ in t u 2 Steffensen Inequality In this section we prove a new ˛-fractional version of Steffensen’s inequality and of Hayashi’s inequality. The results in this and subsequent sections differ from those in [10, 12, 13, 15]. Lemma 1. 0; 1 and a; b 2 R with 0 Ä a < b.