Frontiers in Time Scales and Inequalities

The following sections are included:

• Preface
• Contents

Here we develop the right delta fractional calculus on time scales. This chapter follows [5].

Here we develop the right nabla fractional calculus on time scales. We introduce the related Riemann-Liouville type fractional integral and Caputo like fractional derivative and prove a fractional Taylor formula with integral remainder. It follows [3].

Here we define a Caputo like right discrete delta fractional difference and we produce a right discrete delta fractional Taylor formula for the first time. We estimate the remainder. Then we produce related right discrete delta fractional Ostrowski, Poincaré and Sobolev type inequalities. It follows [3].

Here we define a Caputo like right discrete delta fractional difference and we produce a right discrete delta fractional Taylor formula. We estimate the remainder. Then we produce related right discrete delta fractional Ostrowski, Poincaré and Sobolev type inequalities. It follows [2].

Here we give univariate and multivariate representations of Montgomery type for hybrid functions on Time scales. Based on these we establish univariate and multivariate Ostrowski type inequalities on Time scales domains. These compare the average of a function to its values. The estimates involve the higher order delta and nabla derivatives and partial derivatives. We finish with applications on the time scales ℝ and ℤ. It follows [11].

Here we prove some new type Landau inequalities on Time Scales. Both delta and nabla cases are presented. We give applications. It follows [14].

Here we give univariate and multivariate delta and nabla Grüss type and comparison of means inequalities on Time Scales. The estimates involve the higher order delta and nabla derivatives and partial derivatives of the engaged functions. We finish with applications on the time scales ℝ and ℤ. It follows [11].

Here we present a wide range integral operator general inequalities on time scales under convexity. Our treatment is combined by using the diamond-alpha integral. When that fails in the fractional setting we use the delta and nabla integrals. We give plenty of interesting applications. It follows [6].

Here we obtain a wide range of vectorial integral operator general inequalities on time scales using convexity. Our treatment is combined by using the diamond-alpha integral. When that fails in the fractional setting we employ the delta and nabla integrals. We give many interesting applications. It follows [9].

Using the well known representation formula for functions due to Fink [8], we establish a series of general Grüss and Ostrowski type inequalities involving s-convexity and s-concavity in the second sense, acting to all possible directions. It follows [4].

Using the harmonic polynomials representation formula due to Dedic, Pečaric and Ujevic [6], we establish Ostrowski and Grüss type inequalities involving several functions. The estimates are with respect all norms ‖·‖_p, 1 ≤ p ≤ ∞, and also take into account the s-convexity and s-concavity in the second sense of the involved functions. It follows [3].

Here we present general fractional Hermite-Hadamard type inequalities using m-convexity and (s, m)-convexity. These inequalities are with respect to generalized Riemann-Liouville fractional integrals. Our work is motivated by and expands [8] to the greatest generality and all possible directions. It follows [1].

Here we study generalized fractional integrals and fractional derivatives. We present the reduction method of Fractional Calculus and we reduce them to basic fractional integrals and fractional derivatives. We give a series of generalized Ostrowski type fractional inequalities involving s-convexity. We apply all of the above to Hadamard and Erdélyi-Kober fractional integrals and fractional derivatives. We produce also important generalized fractional Taylor formulae. It follows [4].

The following sections are included:

• Index