Hermite–Hadamard Type Inequalities for Trigonometrically P‑functions

Bekar, Kerim

doi:10.7546/crabs.2019.11.01

Cited by 8 publications

(7 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In ref. [10], Bekar gave the definition of trigonometrically P‐function and related H–H type integral inequalities as follows.Definition A nonnegative Δ : I → ℝ is called trigonometrically P ‐functions if for every ω , ϖ ∈ I and r ∈ [0, 1],

Δ (italicr ω + (1 - r) ϖ) \leq (\sin \frac{π r}{2} + \cos \frac{π r}{2}) [Δ (ω) + Δ (ϖ)] .

Theorem Let the function Δ : [ ω , ϖ ] → ℝ be a trigonometrically P ‐ function . If ω < ϖ and Δ ∈ L [ ω , ϖ ], then the following inequality holds :

\frac{1}{ϖ - ω} \int_{ω}^{ϖ} Δ (x) dx \leq \frac{4}{π} [Δ (ω) + Δ (ϖ)] .

Theorem Let Δ : [ ω , ϖ ] → ℝ, be a trigonometrically P ‐ function .…”

Section: Preliminariesmentioning

confidence: 99%

Geometric trigonometrically convexity and better approximations

Kadakal

2020

Numerical Methods Partial

View full text Add to dashboard Cite

In this manuscript, we introduce the concept of geometric trigonometrically convex functions. We obtain some refinements of the Hermite-Hadamard inequality for functions whose first derivative in absolute value is geometric trigonometrically convex. In addition, it has been proved that the results obtained with Hölder-İşcan and improved power-mean integral inequalities give a better approach than Hölder and power-mean inequalities.

show abstract

Δ (italicr ω + (1 - r) ϖ) \leq (\sin \frac{π r}{2} + \cos \frac{π r}{2}) [Δ (ω) + Δ (ϖ)] .

Theorem Let the function Δ : [ ω , ϖ ] → ℝ be a trigonometrically P ‐ function . If ω < ϖ and Δ ∈ L [ ω , ϖ ], then the following inequality holds :

\frac{1}{ϖ - ω} \int_{ω}^{ϖ} Δ (x) dx \leq \frac{4}{π} [Δ (ω) + Δ (ϖ)] .

Theorem Let Δ : [ ω , ϖ ] → ℝ, be a trigonometrically P ‐ function .…”

Section: Preliminariesmentioning

confidence: 99%

Geometric trigonometrically convexity and better approximations

Kadakal

2020

Numerical Methods Partial

View full text Add to dashboard Cite

show abstract

“…In [3], Bekar obtained the trigonometricallyfunction as follows: In [10], İşcan pointed out the new generalised lemma which is giving many integral inequalities as follows:…”

Section: Definition 3 [23]mentioning

confidence: 99%

Novel Results based on Generalisation of Some Integral Inequalities for Trigonometrically -P Function

Turhan

2020

Sakarya University Journal of Science

View full text Add to dashboard Cite

Trigonometric P-function is defined as a special case of h-convex function. In this article, we used a general lemma that gives trapezoidal, midpoint, Ostrowski, and Simpson type inequalities. With the help of this lemma, we have obtained many integral inequalities and generalisations for trigonometric P-function. We have shown that it goes down to the studies in special cases which are described in our study. Apart from that, we got new results for the trapezoidal, midpoint, Ostrowski, and Simpson type inequalities.

show abstract

“…In [1], Bekar gave the concept of trigonometrically -function as follows:…”

Section: Introductionmentioning

confidence: 99%

“…In [1], Bekar also obtained the following Hermite-Hadamard type inequalities for the trigonometrically -function as follows: Theorem 1. Let the function : [ , ] → ℝ be a trigonometrically -function.…”

Section: Introductionmentioning

confidence: 99%