Some Extended Nabla and Delta Hardy–Copson Type Inequalities with Applications in Oscillation Theory

Hacettepe Journal of Mathematics and Statistics

2022

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In this paper two kinds of dynamic Hardy-Copson type inequalities are derived via diamond alpha integrals. The first kind consists of twelve new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together. The second kind involves another twelve new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Hardy-Copson type inequalities. Our approach is quite new due to the fact that it uses time scale calculus rather than algebra. Therefore both kinds of our results unify some of the known delta and nabla Hardy-Copson type inequalities into one diamond alpha Hardy-Copson type inequalities and offer new Hardy-Copson type inequalities even for the special cases.

Section: Introductionmentioning

confidence: 99%

Diamond alpha Hardy-Copson type dynamic inequalities

Kayar

Hacettepe Journal of Mathematics and Statistics

2022

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“…The growing interest to Hardy-Copson type inequalities take place in the time scale calculus as well and delta unifications of these inequalities are established in the book [4] and in the articles [2,18,44,47,48,[50][51][52][53][54] whereas their nabla counterparts and extensions can be seen in [29][30][31] for ζ > 1.…”

Section: Introductionmentioning

confidence: 99%

The complementary nabla Bennett-Leindler type inequalities

Kayar¹,

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

2022

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We aim to find the complements of the Bennett-Leindler type inequalities in nabla time scale calculus by changing the exponent from $0<\zeta< 1$ to $\zeta>1.$ Different from the literature, the directions of the new inequalities, where $\zeta>1,$ are the same as that of the previous nabla Bennett-Leindler type inequalities obtained for $0<\zeta< 1$. By these settings, we not only complement existing nabla Bennett-Leindler type inequalities but also generalize them by involving more exponents. The dual results for the delta approach and the special cases for the discrete and continuous ones are obtained as well. Some of our results are novel even in the special cases.

“…[35][36][37][38][39][40] For some results about the nabla differential equations and nabla inequalities, see previous works. [41][42][43][44][45][46][47][48][49][50][51] Contrary to delta case, Bennett-Leindler type inequalities had not been considered until the paper 52 appeared. The nabla time scale unifications of the discrete Bennett-Leindler inequalities (9) and (10) and the continuous Bennett-Leindler inequalities ( 13) and ( 14) as well as the nabla versions of Theorems 1-4 for an arbitrary time scale can be seen in the next theorems.…”

Section: Introductionmentioning

confidence: 99%

Diamond alpha Bennett‐Leindler type dynamic inequalities and their applications

Kayar

Math Methods in App Sciences

Pelen

2021

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In this paper, two kinds of dynamic Bennett-Leindler type inequalities via the diamond alpha integrals are derived. The first kind consists of eight new integral inequalities which can be considered as mixed type in the sense that these inequalities contain delta, nabla and diamond alpha integrals together due to the fact that convex linear combinations of delta and nabla Bennett-Leindler type inequalities give diamond alpha Bennett-Leindler type inequalities. The second kind involves four new inequalities, which are composed of only diamond alpha integrals, unifying delta and nabla Bennett-Leindler type inequalities. For the second type, choosing 𝛼 = 1 or 𝛼 = 0 not only yields the same results as the ones obtained for delta and nabla cases but also provides novel results for them.Therefore, both kinds of our results expand some of the known delta and nabla Bennett-Leindler type inequalities, offer new types of these inequalities, and bind and unify them into one diamond alpha Bennett-Leindler type inequalities.Moreover, an application of dynamic Bennett-Leindler type inequalities to the oscillation theory of the second-order half linear dynamic equation is developed and presented for the first time ever.