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A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse...
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A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
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A flexible cable suspended between two points at the same height naturally forms a curve known as a catenary. This shape results from the balance between the cable’s weight and the tension acting along its length, representing a state of mechanical equilibrium. Unlike simpler approximations, the true shape of a hanging cable is described using hyperbolic functions.Hyperbolic functions are closely related to exponential functions and are named for their connection to the geometry of the...
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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...
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Determining the area of a region with straight edges is straightforward, as geometric formulas for rectangles, triangles, and polygons can be applied directly. However, traditional geometric methods are insufficient when a region has a curved boundary, such as the area under a function.fromThe area problem involves finding a systematic way to measure such regions. One approach to solving this problem is through approximation. Instead of attempting to compute the area exactly at the outset, the...
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A region can be enclosed by three curves: a square root function, a reflected cube root function, and a linear function. The linear function intersects each of the other two curves, and these intersection points determine where the boundary of the enclosed region changes. Because different curves serve as the upper and lower boundaries in different parts of the graph, the area cannot be found using a single setup over the entire interval.To compute the area, the region is first divided into two...
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On bounds involving k-Appell's hypergeometric functions.

Muhammad Uzair Awan1, Muhammad Aslam Noor2,3, Marcela V Mihai4

  • 1Department of Mathematics, Government College University, Faisalabad, Pakistan.

Journal of Inequalities and Applications
|June 10, 2017
PubMed
Summary

This study extends the Hermite-Hadamard inequality using k-Riemann-Liouville fractional integrals and k-Appell

Keywords:
convex functionsharmonic convex functionsinequalitiesk-Appell’s hypergeometric functionsk-fractional

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Area of Science:

  • Mathematical Analysis
  • Fractional Calculus
  • Inequalities

Background:

  • The Hermite-Hadamard inequality is a fundamental result in convex analysis.
  • Fractional calculus provides tools for generalizing classical calculus concepts.
  • Harmonic convexity is a specific type of convexity with unique properties.

Purpose of the Study:

  • To derive a new extension of the Hermite-Hadamard inequality using k-Riemann-Liouville fractional integrals.
  • To establish novel k-fractional integral identities.
  • To obtain new k-fractional bounds involving k-Appell's hypergeometric functions for harmonically convex functions.

Main Methods:

  • Derivation of k-fractional integral identities.
  • Application of these identities to establish new inequalities.
  • Utilizing the property of harmonic convexity for the functions involved.

Main Results:

  • A new extension of the Hermite-Hadamard inequality via k-Riemann-Liouville fractional integrals.
  • Two novel k-fractional integral identities.
  • New k-fractional bounds involving k-Appell's hypergeometric functions, serving as k-fractional estimations of trapezoidal and mid-point inequalities.

Conclusions:

  • The derived inequalities offer new insights into fractional integral inequalities for harmonically convex functions.
  • The results generalize and extend existing inequalities in the field.
  • Special cases of the main results are discussed, highlighting their broader applicability.