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A new class of (h, g ; m)-convex functions is presented, together with its properties, thus generalizing several varieties of convexity such as (h−m)-convex functions, (s, m)-Godunova-Levin functions or exponentially (s, m)-convex functions. Also, the Hermite-Hadamard inequality for an (h, g ; m)-convex function is proved and some special results are pointed out.

The recently introduced class of (h, g ; m)-convex functions unifies a certain range of convexity, which allows generalizations of known results. For this class, we present several types of inequality such as Hermite-Hadamard, Fejer and Lah-Ribarič, which generalize and extend corresponding inequalities. We also point out some special refined results.

It was approximately at the same time when lockdown in 2020 started that Erasmus + project mathSTEM: Teaching mathematics in STEM context for STEM students was initiated by five universities from Croatia, North Macedonia and Turkey. Although its intellectual outputs were delayed by the complexity of the Covid situation, this project at the same time gained a new dimension due to this new moment in the world. Here we present how the concept of teaching descriptive geometry fit in the wider frame of mathSTEM teaching and present the contents of the e-platform that was hereby developed. We also give a short overview of LTT events in mathSTEM that were organized for students and for teachers.

Fractional calculus is a theory of differential and integral operators of non-integer order. In recent years, considerable interest in this theory has been stimulated due to its many applications in almost all applied science, especially in numerical analysis and various fields of physics and engineering. Inequalities which involve integrals of functions and their derivatives, whose study has a history of about a century, are of great importance in mathematics, with far-reaching applications in the theory of differential equations, approximations and probability, among others. They occupy a central place in mathematical analysis and its applications. Fractional differentiation inequalities have applications to fractional differential equations ; the most important ones are in establishing uniqueness of the solution of initial problems and giving upper bounds to their solutions. These applications have motivated many researchers in the field of integral inequalities to investigate certain extensions and generalizations using different fractional differential and integral operators. There are several well-known forms of fractional operators: Riemann-Liouville, Weyl, Erdelyi-Kober, Hadamard, Katugampola are just a few. As a solution of fractional order differential or integral equations, the Mittag-Leffler function with its generalizations appears. It is a function of one parameter defined by the power series using the gamma function and it is a natural extension of the exponential, hyperbolic and trigonometric functions. In the book we define the extended and generalized Mittag-Leffler function and give its properties. For different parameter choices, corresponding known generalizations of the Mittag-Leffler function can be deduced: the Wiman generalization, also known as Mittag-Leffler function of two parameters, Prabhakar’s function, Shukla-Prajapati’s function, Salim-Faraj’s function or recent extension defined by Rahman et al. This extended generalized Mittag-Leffler function with the corresponding fractional integral operator (in real domain) is used to obtain fractional generalizations of different types of inequalities, such as inequalities of the Opial, Polya-Szego, Chebyshev, Minkowsky, Hermite, Hadamard and Fejer (for convex, relative convex, m-convex, (h−m)-convex, harmonically convex function, etc). We provide the bounds of various kinds of fractional integral operators containing our extended generalized Mittag-Leffler function and give estimations of these inequalities for different kinds of convex functions. Results are applied on a certain function to establish recurrence relations among Mittag-Leffler functions.

We study a new class of (h, g ; m)-convex functions that unifies a certain range of convexity and allows the generalizations of know results. For this class we present Lah-Ribarič type inequalities from which we obtain inequalities of Hermite- Hadamard, Fejér, Giaccardi, Popoviciu and Petrović.

In this paper, the refinements of integral inequalities for all those types of convex functions are given which can be obtained from (s, m)-convex functions. These inequalities not only provide refinements of bounds for unified integral operators but also for various associated fractional integral operators containing Mittag–Leffler function. At the same time, presented results give generalizations of many known fractional integral inequalities.

This paper presents several fractional generalizations and extensions of known integral inequalities. To obtain these, an extended generalized Mittag-Leffler function and its fractional integral operator are used.

We use Karamata's estimations of the Chebyshev quotient and our extended generalized Mittag-Leffler function with the corresponding fractional integral operator to obtain fractional integral inequalities of Chebyshev type. Also, several known results are improved.

We define an extended generalized Mittag- Leffler function and the corresponding fractional integral operator and use them to obtain generalizations of Opial-type inequalities due to Mitrinović and Pečarić. Also we present certain Polya-Szego type integral inequalities due to Karamata's estimations of the Chebyshev quotient and obtain fractional integral inequalities of Chebyshev type.

Ovaj članak prezentira Beesackov dokaz Opialove nejednakosti te nekoliko jednostavnijih dokaza koji su netom uslijedili. Pritom se posebno promatra Beesackova metoda dokaza koja je doprinijela razvoju općenitijih nejednakosti Opailova tipa.

We consider generalizations of Opial's inequality due to Willet, Godunova, Levin and Rozanova in one variable and obtain certain improvements. Then we extend Willet's and Rozanova's Opial-type inequalities to inequalities in several independent variables. Also, we present some new Opial-type inequalities in several independent variables.

We present extensions of Opial-type inequalities and use them to obtain a generalization of an inequality due to Mitrinović and Pečarić for relatively convex functions. Also, some new Opial-type inequalities are given involving different types of fractional integrals and fractional derivatives as applications.

The purpose of this corrigendum is to correct an error in the earlier paper by the authors: Generalizations of Opial-type inequalities in several independent variables, Demonstratio Math.

Ovaj članak prezentira Opialovu nejednakost i njen originalni dokaz. Također je promatran Olechov dokaz Opialove nejednakosti te njegov doprinos oslabljenju prvotnih uvjeta.

Over the last five decades, an enormous amount of work has been done on Opial's integral inequality, dealing with new proofs, various generalizations, extensions and discrete analogues. The Opial inequality is recognized as fundamental result in the analysis of qualitative properties of solution of differential equations. We use submultiplicative convex functions, appropriate representations of functions and inequalities involving means to obtain generalizations and extensions of certain known multidimensional integral and discrete Opial-type inequalities.

Knjiga sadrži apstrakte svih predavanja održanih tijekom konferencije Mathematical Inequalities and Applications 2015, International conference organized on the occasion of 60th birthdays of Professors Neven Elezović, Marko Matić and Ivan Perić, Mostar, Bosna i Hercegovina, November 11- 15, 2015.

In 1960, the Polish mathematician Z. Opial proved integral inequality, which is recognized as a fundamental result in the analysis of qualitative properties of a solution of differential equations. Over the last five decades, an enormous amount of work has been done on the Opial inequality: several simplifications of the original proof, various extensions, generalizations and discrete analogues. Motivated with Opial-type inequalities, together with Jensen's inequality, we improve some known results and obtain new, interesting inequalities. For such inequalities we construct functionals and give its mean value theorems. These Cauchy type mean value theorems are used for Stolarsky type means, all defined by the observed inequalities, and also, they are used to prove the n-exponential convexity for the functionals. We study Opial--type inequalities not only for ordinary derivatives, but also for fractional derivatives which leads us to the fractional calculus. It is a theory of differential and integral operators of non-integer order that has become very useful due to its many applications in almost all the applied sciences. We study the Riemann-Liouville fractional integrals and three types of fractional derivatives (the Riemann-Liouville, the Caputo and the Canavati type), in the real domain. Obtaining improvements of composition identities for the above mentioned fractional derivatives, we apply them on the fractional differentiation inequalities that have applications in the fractional differential equations ; the most important ones are in establishing the uniqueness of the solution of initial problems and giving upper bounds to their solutions. We give refinements, generate new extensions and generalizations of some known Opial-type inequalities, investigate the possibility of obtaining the best possible constant, compare results obtained by different methods and present some new inequalities involving fractional integrals and fractional derivatives.

Knjiga sadrži abstrakte svih predavanja održanih tijekom konferencije Mathematical Inequalities and Applications 2014, One Thousand Papers Conference, Trogir, June 22-26, 2014.

In this paper we consider generalizations of Opial's inequality due to Willett, Godunova, Levin and Rozanova. Cauchy type mean value theorems are proved and used in studying Stolarsky type means defined by the obtained inequalities. Also, a method of producing n-exponentially convex and exponentially convex functions is applied.

In this paper we give extensions of Opial-type integral inequalities and use them to obtain a generalization of an inequality due to Agarwal and Pang. Applications with respect to fractional derivatives and fractional integrals are also given.

In this paper we consider a certain class of convex functions in Opial-type integral inequalities. Cauchy type mean value theorems are proved and used in studying Stolarsky type means defined by the observed integral inequalities. A method of producing n- exponentially convex and exponentially convex functions is applied. Also, some new Opial-type equalities are given involving fractional integrals and fractional derivatives.

Some known Opial-type inequalities are extended and used to obtained inequalities for q-convex functions. A fractional versions of these inequalities are also given, involving the Riemann-Liouville fractional integrals and the Caputo fractional derivatives.

Using the Laplace transform we prove that some conditions in composition identities for the left-sided and right-sided Caputo fractional derivatives can be relaxed. As an application, an Opial-type inequality involving the Caputo fractional derivatives is presented.

Using the Laplace transform we prove that some conditions in the composition rule for the Canavati fractional derivatives can be relaxed. As an application, an Opial-type inequality involving the Canavati fractional derivatives is presented.

U disertaciji su promatrani Riemann-Liouvilleovi razlomljeni integrali i tri tipa razlomljenih derivacija (Riemann-Liouvilleov, Caputov i Canavatijev tip), te nejednakosti koje uključuju navedene razlomljene derivacije. Nakon prvog poglavlja u kojem je dan pregled osnovnih pojmova i rezultata potrebnih u radu, slijedi poglavlje o kompozicijskim identitetima za razlomljene derivacije. Prikazani su rezultati poboljšanja postojećih kompozicijskih identiteta. Posebno su dani kompozicijski identiteti za svaki od tri tipa razlomljenih derivacija. Dan je pregled uvjeta pod kojima kompozicijski identiteti vrijede. Pritom se vodilo računa o ulozi početnih uvjeta za funkciju uključenu u kompozicijske identitete te o međusobnom odnosu redova Riemann-Liouvilleovog razlomljenog integrala i navedenih razlomljenih derivacija. Ovako dobiveni rezultati primijenjeni su na razlomljene diferencijalne nejednakosti u zadnjem poglavlju. Proučavane su nejednakosti Opialovog tipa, te su dani rezultati i nejednakosti koje uključuju Riemann-Liouvilleove, Caputove i Canavatijeve razlomljene derivacije. Dana su poopćenja i proširenja Opialove nejednakosti, zatim neka profinjenja te su dobivene nove razlomljene diferencijalne nejednakosti. Istražene su mogućnosti dobivanja najboljih mogućih konstanti i izvršena je usporedba rezultata dobivenih različitim metodama. Naglašeni su posebni slučajevi redova razlomljenih derivacija, kada se nejednakosti svode na klasičnu Opialovu, Beesackovu, Wirtingerovu, Finkovu, Agarwal-Pangovu ili Alzerove verziju Opialove nejednakosti.