By Panos M. Pardalos, Themistocles M. Rassias
The contributions during this quantity goal to deepen understanding of many of the present examine difficulties and theories in glossy subject matters akin to calculus of diversifications, optimization idea, advanced research, actual research, differential equations, and geometry. Applications to those parts of arithmetic are provided inside of the broad spectrum of analysis in Engineering technology with particular emphasis on equilibrium difficulties, complexity in numerical optimization, dynamical structures, non-smooth optimization, advanced community research, statistical types and information mining, and effort structures. extra emphasis is given to interdisciplinary examine, even though matters are handled in a unified and self-contained demeanour. The presentation of tools, idea and purposes makes this tribute a useful reference for academics, researchers, and different pros drawn to pure and utilized examine, philosophy of arithmetic, and arithmetic education. Some overview papers released during this quantity can be relatively important for a broader viewers of readers in addition to for graduate scholars who look for the newest information.
Constantin Carathéodory’s wide-ranging impact within the foreign mathematical neighborhood was once noticeable throughout the first Fields Medals awards on the foreign Congress of Mathematicians, Oslo, 1936. medals have been provided, one to Lars V. Ahlfors and one to Jesse Douglass. It was once Carathéodory who awarded either their works through the starting of the overseas Congress. This quantity comprises major papers in technology and Engineering devoted to the reminiscence of Constantin Carathéodory and the spirit of his mathematical impact.
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Theorem 9 (Jensen Inequality). 0; 1 and a; b; x; y 2 Œ0; 1/. If w W R ! R and g W R ! x; y/ ! R is continuous and convex, then ! t/d˛ t Proof. 3] and thus is omitted. t u Theorem 10 (Grüss Inequality). Let a; b; s 2 Œ0; 1/, and let f ; g W Œa; b ! R be continuous functions. t/d˛ tˇˇ m2 /: Proof. t/. M1 C m1 /2 m1 /2 : Now consider the case: r WD Z ˛ b˛ a˛ where r 2 R. R. Anderson Let us now turn to the case involving general functions f and g under assumptions (16). t/d˛ t a and the earlier cases, one can easily finish the proof as in the case with ˛ D 1.
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 t 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  ˛ 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.
Some new scales of refined Hardy type inequalities via functions related to superquadracity. Math. Inequal. Appl. 16, 679–695 (2013) 2. : Some new refined Hardy type inequalities with breaking points p D 2 or p D 3. In: Proceedings of the IWOTA 2011, Operator Theory: Advances and Applications, vol. 236, pp. 1–10. Birkhäuser/Springer, Basel/Berlin (2014) 3. : Inequalities for averages of quasiconvex and superquadratic functions. Math. Inequal. Appl. 19(2), 535–550 (2016) 4. : Inequalities for averages of convex and superquadratic functions.