Generalized fractional midpoint type inequalities for co-ordinated convex functions

In this research paper, we investigate generalized fractional integrals to obtain midpoint type inequalities for the co-ordinated convex functions. First of all, we establish an identity for twice partially differentiable mappings. By utilizing this equality, some midpoint type inequalities via generalized fractional integrals are proved. We also show that the main results reduce some midpoint inequalities given in earlier works for Riemann integrals and Riemann-Liouville fractional integrals. Finally, some new inequalities for $k$-Riemann-Liouville fractional integrals are presented as special cases of our results.

Some Integral Inequalities for Generalized Convex Fuzzy-Interval-Valued Functions via Fuzzy Riemann Integrals

In this study, we introduce the new concept of $$h$$ h -convex fuzzy-interval-valued functions. Under the new concept, we present new versions of Hermite–Hadamard inequalities (H–H inequalities) are called fuzzy-interval Hermite–Hadamard type inequalities for $$h$$ h -convex fuzzy-interval-valued functions ($$h$$ h -convex FIVF) by means of fuzzy order relation. This fuzzy order relation is defined level wise through Kulisch–Miranker order relation defined on fuzzy-interval space. Fuzzy order relation and inclusion relation are two different concepts. With the help of fuzzy order relation, we also present some H–H type inequalities for the product of $$h$$ h -convex FIVFs. Moreover, we have also established strong relationship between Hermite–Hadamard–Fej´er (H–H–Fej´er) type inequality and $$h$$ h -convex FIVF. There are also some special cases presented that can be considered applications. There are useful examples provided to demonstrate the applicability of the concepts proposed in this study. This paper's thoughts and methodologies could serve as a springboard for more research in this field.

Fuzzy-interval inequalities for generalized convex fuzzy-interval-valued functions via fuzzy Riemann integrals

<abstract> <p>The objective of the authors is to introduce the new class of convex fuzzy-interval-valued functions (convex-FIVFs), which is known as $ p $-convex fuzzy-interval-valued functions ($ p $-convex-FIVFs). Some of the basic properties of the proposed fuzzy-interval-valued functions are also studied. With the help of $ p $-convex FIVFs, we have presented some Hermite-Hadamard type inequalities ($ H-H $ type inequalities), where the integrands are FIVFs. Moreover, we have also proved the Hermite-Hadamard-Fejér type inequality ($ H-H $ Fejér type inequality) for $ p $-convex-FIVFs. To prove the validity of main results, we have provided some useful examples. We have also established some discrete form of Jense's type inequality and Schur's type inequality for $ p $-convex-FIVFs. The outcomes of this paper are generalizations and refinements of different results which are proved in literature. These results and different approaches may open new direction for fuzzy optimization problems, modeling, and interval-valued functions.</p> </abstract>

Some fuzzy-interval integral inequalities for harmonically convex fuzzy-interval-valued functions

<abstract> <p>It is well-known fact that fuzzy interval-valued functions (F-I-V-Fs) are generalizations of interval-valued functions (I-V-Fs), and inclusion relation and fuzzy order relation on interval space and fuzzy space are two different concepts. Therefore, by using fuzzy order relation (FOR), we derive inequalities of Hermite-Hadamard (<italic>H</italic>·<italic>H</italic>) and Hermite-Hadamard Fejér (<italic>H</italic>·<italic>H</italic> Fejér) like for harmonically convex fuzzy interval-valued functions by applying fuzzy Riemann integrals. Moreover, we establish the relation between fuzzy integral inequalities and fuzzy products of harmonically convex fuzzy interval-valued functions. The outcomes of this study are generalizations of many known results which can be viewed as an application of a defined new version of inequalities.</p> </abstract>

System analysis of basic of mechanics

Механика берет свое начало со статики. Основным понятием статики является понятие «сила». При нарушении равновесия возникает движение, которое определяется скоростью и ускорением в координатной системе пространство–время; скорость определяется как отношение мгновенного изменения координаты к соответствующему мгновенному изменению времени. В свою очередь изменение мгновенной скорости, т. е. ускорение, связано с воздействием силы за мгновенное время, и называется импульсом силы. Второй закон Ньютона как основной закон динамики сформулирован для воздействия на тело постоянной силы за короткий промежуток времени, т. е. импульса силы. Импульс силы вызывает изменение скорости движения тела; мерой сопротивления тела изменениям скорости является масса; произведением массы на скорость вводится понятие «количество движения» (импульс). Поэтому второй закон Ньютона определяет силу как отношение изменения количества движения к короткому времени действия импульса силы. Короткое время действия силы является частным случаем непрерывного ее действия во времени. В данном исследовании импульс силы понимается в обобщенном представлении как произведение силы на непрерывное время действия. По аналогии импульсу силы во времени вводится импульс силы в пространстве. С позиции системного анализа графиков сила–время, масса–скорость, сила–пространство, мощность время построены дифференциальные и интегральные законы динамики потенциально связного взаимодействия соответственно сила–время–масса– скорость, сила–пространство–работа, мощность–время–энергия. Анализ полных дифференциалов потенциалов приводит к представлениям функционального времени и пространства, которые сопряжено дополняют время и пространство взаимодействия. Время и пространство действия силы в исследуемых системах по аналогии с массой рассматриваются как меры сопротивления тела изменениям силы, т. е. как механические параметры, а не геометрические. Интегральные законы динамики построены в виде суперпозиции интегралов Римана для прямых функций и интегралов Стилтьеса для обратных. Интегралы Римана описывают современную динамику, а интегралы Стилтьеса ее дополнение до потенциальной. Mechanics starts with statics. The main concept of statics is the concept of force. When the equilibrium is disturbed, motion occurs, which is determined by the speed and acceleration in the space-time coordinate system; speed is defined as the ratio of an instantaneous change in the coordinate to the corresponding instantaneous change in time. In turn, the change in instantaneous speed, i.e. acceleration, is associated with the impact of a force in an instantaneous time, which is called the force pulse. The second law of Newton, as the basic law of dynamics, is formulated for the effect on the body of a constant force for a short period of time, i.e., the force impulse. The force pulse causes a change in the speed of the body; the measure of the body's resistance to changes in speed is the mass; the product of mass and speed is introduced the concept of the amount of movement (momentum). Therefore, Newton's second law defines force as the ratio of the change in the amount of motion to the short time of action of the force impulse. The short duration of the force is a special case of continuous time. In this study, the force impulse is understood in a generalized representation as the product of the force for a continuous time of action. By analogy with a force pulse in time, a force pulse in space is introduced. With the system chart analysis force-time, mass speed, force, space, power is the differential and integral laws of dynamics potentially Svyaznoy interaction, respectively, the power–time–weight–speed, power– space–work, power–time–energy. The analysis of complete potential differentials leads to representations of functional time and space that complement the interaction time and space. The time and space of the force action in the studied systems are considered by analogy with mass as measures of the body's resistance to changes in force, i. e. as mechanical parameters, rather than geometric ones. The integral laws of dynamics are constructed as a superposition of Riemann integrals for direct functions and stiltjes integrals for inverse functions. Riemann integrals describe modern dynamics, and stiltjes integrals describe its complement to the potential one.

A New Version of the Hermite–Hadamard Inequality for Riemann–Liouville Fractional Integrals

Integral inequalities play a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods. Thus, the present days need to seek accurate inequalities for proving the existence and uniqueness of the mathematical methods. The concept of convexity plays a strong role in the field of inequalities due to the behavior of its definition. There is a strong relationship between convexity and symmetry. Whichever one we work on, we can apply it to the other one due the strong correlation produced between them, especially in the past few years. In this article, we firstly point out the known Hermite–Hadamard (HH) type inequalities which are related to our main findings. In view of these, we obtain a new inequality of Hermite–Hadamard type for Riemann–Liouville fractional integrals. In addition, we establish a few inequalities of Hermite–Hadamard type for the Riemann integrals and Riemann–Liouville fractional integrals. Finally, three examples are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.

On the Generalized Hermite–Hadamard Inequalities via the Tempered Fractional Integrals

Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of λ -incomplete gamma function. Using the new notation, we established a few inequalities of the Hermite–Hadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, Riemann–Liouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.