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Journal articles on the topic 'Convexity estimates'

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Published: 10 December 2022

Last updated: 28 January 2023

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1

Shi, Shujun. "Convexity estimates for the Green’s function." Calculus of Variations and Partial Differential Equations 53, no. 3-4 (August 12, 2014): 675–88. http://dx.doi.org/10.1007/s00526-014-0763-4.

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2

Grunau, Hans-Christoph, and Stephan Lenor. "Uniform estimates and convexity in capillary surfaces." Nonlinear Analysis: Theory, Methods & Applications 97 (March 2014): 83–93. http://dx.doi.org/10.1016/j.na.2013.11.016.

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3

Almutairi, Ohud Bulayhan. "Quantum Estimates for Different Type Intequalities through Generalized Convexity." Entropy 24, no. 5 (May 20, 2022): 728. http://dx.doi.org/10.3390/e24050728.

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This article estimates several integral inequalities involving (h−m)-convexity via the quantum calculus, through which Important integral inequalities including Simpson-like, midpoint-like, averaged midpoint-trapezoid-like and trapezoid-like are extended. We generalized some quantum integral inequalities for q-differentiable (h−m)-convexity. Our results could serve as the refinement and the unification of some classical results existing in the literature by taking the limit q→1−.

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4

Fraser, Ailana. "Index estimates for minimal surfaces and $k$-convexity." Proceedings of the American Mathematical Society 135, no. 11 (November 1, 2007): 3733–45. http://dx.doi.org/10.1090/s0002-9939-07-08894-6.

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5

Andrews, Ben, Mat Langford, and James McCoy. "Convexity estimates for surfaces moving by curvature functions." Journal of Differential Geometry 99, no. 1 (January 2015): 47–75. http://dx.doi.org/10.4310/jdg/1418345537.

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6

Alessandroni, Roberta, and Carlo Sinestrari. "Convexity estimates for a nonhomogeneous mean curvature flow." Mathematische Zeitschrift 266, no. 1 (June 5, 2009): 65–82. http://dx.doi.org/10.1007/s00209-009-0554-3.

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7

Andrews, Ben, Mathew Langford, and James McCoy. "Convexity estimates for hypersurfaces moving by convex curvature functions." Analysis & PDE 7, no. 2 (May 30, 2014): 407–33. http://dx.doi.org/10.2140/apde.2014.7.407.

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8

Cao, Jia-Ding, and Heinz H. Gonska. "Pointwise estimates for higher order convexity preserving polynomial approximation." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 36, no. 2 (October 1994): 213–33. http://dx.doi.org/10.1017/s0334270000010365.

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AbstractDeVore-Gopengauz-type operators have attracted some interest over the recent years. Here we investigate their relationship to shape preservation. We construct certain positive convolution-type operators Hn, s, j which leave the cones of j-convex functions invariant and give Timan-type inequalities for these. We also consider Boolean sum modifications of the operators Hn, s, j show that they basically have the same shape preservation behavior while interpolating at the endpoints of [−1, 1], and also satisfy Telyakovskiῐ- and DeVore-Gopengauz-type inequalities involving the first and second order moduli of continuity, respectively. Our results thus generalize related results by Lorentz and Zeller, Shvedov, Beatson, DeVore, Yu and Leviatan.

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9

Garofalo, Nicola. "Geometric second derivative estimates in Carnot groups and convexity." manuscripta mathematica 126, no. 3 (March 26, 2008): 353–73. http://dx.doi.org/10.1007/s00229-008-0182-y.

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10

Edelen, Nick. "Convexity estimates for mean curvature flow with free boundary." Advances in Mathematics 294 (May 2016): 1–36. http://dx.doi.org/10.1016/j.aim.2016.02.026.

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11

Atshan, Waggas Galib, and Ali Hamza Abada. "On Subclass of -Uniformly Convex Functions of Complex Order Involving Multiplier Transformations." Abstract and Applied Analysis 2012 (2012): 1–10. http://dx.doi.org/10.1155/2012/150571.

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We introduce a subclass of -uniformly convex functions of order with negative coefficients by using the multiplier transformations in the open unit disk . We obtain coefficient estimates, radii of convexity and close-to-convexity, extreme points, and integral means inequalities for the function that belongs to the class .

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12

ZLATANOV, BOYAN. "Error estimates for approximating best proximity points for cyclic contractive maps." Carpathian Journal of Mathematics 32, no. 2 (2016): 265–70. http://dx.doi.org/10.37193/cjm.2016.02.15.

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We find a priori and a posteriori error estimates of the best proximity point for the Picard iteration associated to a cyclic contraction map, which is defined on a uniformly convex Banach space with modulus of convexity of power type.

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13

Ma, Xi-Nan, Shujun Shi, and Yu Ye. "The Convexity Estimates for the Solutions of Two Elliptic Equations." Communications in Partial Differential Equations 37, no. 12 (January 2012): 2116–37. http://dx.doi.org/10.1080/03605302.2012.727129.

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14

Guessab, Allal, and Gerhard Schmeisser. "Convexity results and sharp error estimates in approximate multivariate integration." Mathematics of Computation 73, no. 247 (December 19, 2003): 1365–85. http://dx.doi.org/10.1090/s0025-5718-03-01622-3.

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15

Cellina, A. "Comparison Results and Estimates on the Gradient without Strict Convexity." SIAM Journal on Control and Optimization 46, no. 2 (January 2007): 738–49. http://dx.doi.org/10.1137/060655869.

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16

Rashid, Saima, Muhammad Aslam Noor, and Khalida Inayat Noor. "New Estimates for Exponentially Convex Functions via Conformable Fractional Operator." Fractal and Fractional 3, no. 2 (April 15, 2019): 19. http://dx.doi.org/10.3390/fractalfract3020019.

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In this paper, we derive a new Hermite–Hadamard inequality for exponentially convex functions via α -fractional integral. We also prove a new integral identity. Using this identity, we establish several Hermite–Hadamard type inequalities for exponentially convexity, which can be obtained from our results. Some special cases are also discussed.

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17

Merad, Meriem, Badreddine Meftah, and Abdourazek Souahi. "Integral inequalities via harmonically h-convexity." Moroccan Journal of Pure and Applied Analysis 7, no. 3 (March 30, 2021): 385–99. http://dx.doi.org/10.2478/mjpaa-2021-0026.

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Abstract In this paper, we establish some estimates of the left side of the generalized Gauss-Jacobi quadrature formula for harmonic h-preinvex functions involving Euler’s beta and hypergeometric functions. The obtained results are mainly based on the identity given by M. A. Noor, K. I. Noor and S. Iftikhar in [17].

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18

Deniz, Erhan, Yücel Özkan, and Luminiţa-Ioana Cotîrlă. "Subclasses of Uniformly Convex Functions with Negative Coefficients Based on Deniz–Özkan Differential Operator." Axioms 11, no. 12 (December 14, 2022): 731. http://dx.doi.org/10.3390/axioms11120731.

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We introduce in this paper a new family of uniformly convex functions related to the Deniz–Özkan differential operator. By using this family of functions with a negative coefficient, we obtain coefficient estimates, the radius of starlikeness, convexity, and close-to-convexity, and we find their extreme points. Moreover, the neighborhood, partial sums, and integral means of functions for this new family are studied.

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19

Alimohammadi, Davood, Nak Eun Cho, Ebrahim Analouei Adegani, and Ahmad Motamednezhad. "Argument and Coefficient Estimates for Certain Analytic Functions." Mathematics 8, no. 1 (January 5, 2020): 88. http://dx.doi.org/10.3390/math8010088.

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The aim of the present paper is to introduce a new class G α , δ of analytic functions in the open unit disk and to study some properties associated with strong starlikeness and close-to-convexity for the class G α , δ . We also consider sharp bounds of logarithmic coefficients and Fekete-Szegö functionals belonging to the class G α , δ . Moreover, we provide some topics related to the results reported here that are relevant to outcomes presented in earlier research.

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20

AOUF, M. K., A. SHAMANDY, and M. F. YASSEN. "ON CERTAIN SUBCLASS OF UNIVALENT FUNCTIONS IN THE UNIT DISC I." Tamkang Journal of Mathematics 26, no. 4 (December 1, 1995): 299–312. http://dx.doi.org/10.5556/j.tkjm.26.1995.4409.

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The object of the present paper is to derive several interesting proper- ties of the class $P_n(\alpha, \beta, \gamma)$ consisting of analytic and univalent functions with neg- ative coefficients. Coefficient estimates, distortion theorems and closure theorems of functions in the class $P_n(\alpha, \beta, \gamma)$ are determined. Also radii of close-to-convexity, starlikeness and convexity and integral operators are determined.

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21

Ali Shah, Syed Ghoos, Saqib Hussain, Saima Noor, Maslina Darus, and Ibrar Ahmad. "Multivalent Functions Related with an Integral Operator." International Journal of Mathematics and Mathematical Sciences 2021 (December 6, 2021): 1–13. http://dx.doi.org/10.1155/2021/5882343.

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In this present paper, we introduce and explore certain new classes of uniformly convex and starlike functions related to the Liu–Owa integral operator. We explore various properties and characteristics, such as coefficient estimates, rate of growth, distortion result, radii of close-to-convexity, starlikeness, convexity, and Hadamard product. It is important to mention that our results are a generalization of the number of existing results in the literature.

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22

Murugusundaramoorthy, G., and N. Magesh. "Certain Subclasses of Starlike Functions of Complex Order Involving Generalized Hypergeometric Functions." International Journal of Mathematics and Mathematical Sciences 2010 (2010): 1–12. http://dx.doi.org/10.1155/2010/178605.

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Making use of the generalized hypergeometric functions, we define a new subclass of uniformly convex functions and a corresponding subclass of starlike functions with negative coefficients and obtain coefficient estimates, extreme points, the radii of close-to-convexity, starlikeness and convexity, and neighborhood results for the classTSml(α,β,γ). In particular, we obtain integral means inequalities for the functionfthat belongs to the classTSml(α,β,γ)in the unit disc.

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23

Xu, Shicheng. "Local estimate on convexity radius and decay of injectivity radius in a Riemannian manifold." Communications in Contemporary Mathematics 20, no. 06 (August 27, 2018): 1750060. http://dx.doi.org/10.1142/s0219199717500602.

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In this paper, we prove the following pointwise and curvature-free estimates on convexity radius, injectivity radius and local behavior of geodesics in a complete Riemannian manifold [Formula: see text]: (1) the convexity radius of [Formula: see text], [Formula: see text], where [Formula: see text] is the injectivity radius of [Formula: see text] and [Formula: see text] is the focal radius of open ball centered at [Formula: see text] with radius [Formula: see text]; (2) for any two points [Formula: see text] in [Formula: see text], [Formula: see text] where [Formula: see text] is the conjugate radius of [Formula: see text]; (3) for any [Formula: see text], any (not necessarily minimizing) geodesic in [Formula: see text] has length [Formula: see text]. We also clarify two different concepts on convexity radius and give examples to illustrate that the one more frequently used in literature is not continuous.

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24

El-Ashwah, R. M., M. K. Aouf, A. A. M. Hassan, and A. H. Hassan. "A New Class of Analytic Functions Defined by Using Salagean Operator." International Journal of Analysis 2013 (February 5, 2013): 1–10. http://dx.doi.org/10.1155/2013/153128.

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We derive some results for a new class of analytic functions defined by using Salagean operator. We give some properties of functions in this class and obtain numerous sharp results including for example, coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, integral means inequalities, and partial sums of functions belonging to this class. Finally, we give an application involving certain fractional calculus operators that are also considered.

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25

Ihsan Butt, Saad, Hüseyin Budak, and Kamsing Nonlaopon. "New Quantum Mercer Estimates of Simpson–Newton-like Inequalities via Convexity." Symmetry 14, no. 9 (September 16, 2022): 1935. http://dx.doi.org/10.3390/sym14091935.

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Recently, developments and extensions of quadrature inequalities in quantum calculus have been extensively studied. As a result, several quantum extensions of Simpson’s and Newton’s estimates are examined in order to explore different directions in quantum studies. The main motivation of this article is the development of variants of Simpson–Newton-like inequalities by employing Mercer’s convexity in the context of quantum calculus. The results also give new quantum bounds for Simpson–Newton-like inequalities through Hölder’s inequality and the power mean inequality by employing the Mercer scheme. The validity of our main results is justified by providing examples with graphical representations thereof. The obtained results recapture the discoveries of numerous authors in quantum and classical calculus. Hence, the results of these inequalities lead us to the development of new perspectives and extensions of prior results.

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26

Nguyen, Huy The. "Convexity and cylindrical estimates for mean curvature flow in the sphere." Transactions of the American Mathematical Society 367, no. 7 (March 4, 2015): 4517–36. http://dx.doi.org/10.1090/s0002-9947-2015-05927-3.

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27

Imbert, Cyril. "Convexity of solutions and C1,1 estimates for fully nonlinear elliptic equations." Journal de Mathématiques Pures et Appliquées 85, no. 6 (June 2006): 791–807. http://dx.doi.org/10.1016/j.matpur.2006.01.003.

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28

Gartland, Chris. "Estimates on the Markov convexity of Carnot groups and quantitative nonembeddability." Journal of Functional Analysis 279, no. 8 (November 2020): 108697. http://dx.doi.org/10.1016/j.jfa.2020.108697.

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29

Kalsoom, Humaira, Muhammad Amer, Moin-ud-Din Junjua, Sabir Hussain, and Gullnaz Shahzadi. "Some (p,q)-Estimates of Hermite-Hadamard-Type Inequalities for Coordinated Convex and Quasi- Convex Functions." Mathematics 7, no. 8 (July 31, 2019): 683. http://dx.doi.org/10.3390/math7080683.

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In this paper, we present the preliminaries of ( p , q ) -calculus for functions of two variables. Furthermore, we prove some new Hermite-Hadamard integral-type inequalities for convex functions on coordinates over [ a , b ] × [ c , d ] by using the ( p , q ) -calculus of the functions of two variables. Furthermore, we establish an identity for the right-hand side of the Hermite-Hadamard-type inequalities on coordinates that is proven by using the ( p , q ) -calculus of the functions of two variables. Finally, we use the new identity to prove some trapezoidal-type inequalities with the assumptions of convexity and quasi-convexity on coordinates of the absolute values of the partial derivatives defined in the ( p , q ) -calculus of the functions of two variables.

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30

Qi, Hengxiao, Waqas Nazeer, Fatima Abbas, and Wenbo Liao. "Some Inequalities of Hermite–Hadamard Type for MT-h-Convex Functions via Classical and Generalized Fractional Integrals." Journal of Function Spaces 2022 (July 16, 2022): 1–9. http://dx.doi.org/10.1155/2022/1257104.

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Convexity plays a vital role in pure and applied mathematics specially in optimization theory, but the classical convexity is not enough to fulfil the needs of modern mathematics; hence, it is important to study generalized notion of convexity. Fraction integral operators also become an important tool for solving problems of model physical and engineering processes that are found to be best described by fractional differential equations. The aim of this paper is to study MT-h-convex functions via fractional integral operators. We establish several Hermite–Hadamard-type inequalities for MT-h-convex function via classical and generalized fractional integrals. We also obtain special means related to our results and present some error estimates for the trapezoidal formulas.

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31

Nasir, Jamshed, Shahid Qaisar, Saad Ihsan Butt, Hassen Aydi, and Manuel De la Sen. "Hermite-Hadamard like inequalities for fractional integral operator via convexity and quasi-convexity with their applications." AIMS Mathematics 7, no. 3 (2021): 3418–39. http://dx.doi.org/10.3934/math.2022190.

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<abstract><p>Since the supposed Hermite-Hadamard inequality for a convex function was discussed, its expansions, refinements, and variations, which are called Hermite-Hadamard type inequalities, have been widely explored. The main objective of this article is to acquire new Hermite-Hadamard type inequalities employing the Riemann-Liouville fractional operator for functions whose third derivatives of absolute values are convex and quasi-convex in nature. Some special cases of the newly presented results are discussed as well. As applications, several estimates concerning Bessel functions and special means of real numbers are illustrated.</p></abstract>

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32

Anastassiou, George A. "Multivariate and abstract approximation theory for Banach space valued functions." Demonstratio Mathematica 50, no. 1 (August 28, 2017): 208–22. http://dx.doi.org/10.1515/dema-2017-0020.

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Abstract Here we study quantitatively the high degree of approximation of sequences of linear operators acting on Banach space valued Fréchet differentiable functions to the unit operator, as well as other basic approximations including those under convexity. These operators are bounded by real positive linear companion operators. The Banach spaces considered here are general and no positivity assumption is made on the initial linear operators for which we study their approximation properties. We derive pointwise and uniform estimates, which imply the approximation of these operators to the unit assuming Fréchet differentiability of functions, and then we continue with basic approximations. At the end we study the special case where the approximated function fulfills a convexity condition resulting into sharp estimates. We give applications to Bernstein operators.

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33

Naeem, Muhammad, Saqib Hussain, Tahir Mahmood, Shahid Khan, and Maslina Darus. "A New Subclass of Analytic Functions Defined by Using Salagean q-Differential Operator." Mathematics 7, no. 5 (May 21, 2019): 458. http://dx.doi.org/10.3390/math7050458.

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In our present investigation, we use the technique of convolution and quantum calculus to study the Salagean q-differential operator. By using this operator and the concept of the Janowski function, we define certain new classes of analytic functions. Some properties of these classes are discussed, and numerous sharp results such as coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, and integral mean inequalities of functions belonging to these classes are obtained and studied.

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34

Mahmood, Zainab H., Buthyna N. Shihab, and Kassim A. Jassim. "Certain Family of Multivalent Functions Associated With Subordination." Ibn AL- Haitham Journal For Pure and Applied Sciences 33, no. 1 (January 20, 2020): 96. http://dx.doi.org/10.30526/33.1.2377.

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The main objectives of this pepper are to introduce new classes. We have attempted to obtain coefficient estimates, radius of convexity, Distortion and Growth theorem and other related results for the classes

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35

Meftah, B., M. Benssaad, W. Kaidouchi, and S. Ghomrani. "Conformable fractional Hermite-Hadamard type inequalities for product of two harmonic 𝑠-convex functions." Proceedings of the American Mathematical Society 149, no. 4 (February 5, 2021): 1495–506. http://dx.doi.org/10.1090/proc/15396.

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In this paper, we establish some conformable fractional Hermite-Hadamard type integral inequalities via harmonic s s -convexity, and the estimates of the products of two harmonic s s -convex functions are also considered.

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36

Alber, Y. I., R. S. Burachik, and A. N. Iusem. "A proximal point method for nonsmooth convex optimization problems in Banach spaces." Abstract and Applied Analysis 2, no. 1-2 (1997): 97–120. http://dx.doi.org/10.1155/s1085337597000298.

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In this paper we show the weak convergence and stability of the proximal point method when applied to the constrained convex optimization problem in uniformly convex and uniformly smooth Banach spaces. In addition, we establish a nonasymptotic estimate of convergence rate of the sequence of functional values for the unconstrained case. This estimate depends on a geometric characteristic of the dual Banach space, namely its modulus of convexity. We apply a new technique which includes Banach space geometry, estimates of duality mappings, nonstandard Lyapunov functionals and generalized projection operators in Banach spaces.

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37

Han, Haeyoon, Hanik Kim, and Hyochoong Bang. "Monocular Pose Estimation of an Uncooperative Spacecraft Using Convexity Defect Features." Sensors 22, no. 21 (November 6, 2022): 8541. http://dx.doi.org/10.3390/s22218541.

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Spacecraft relative pose estimation for an uncooperative spacecraft is challenging because the target spacecraft neither provides sensor information to a chaser spacecraft nor contains markers that assist vision-based navigation. Moreover, the chaser does not have prior pose estimates when initiating the pose estimation. This paper proposes a new monocular pose estimation algorithm that addresses these issues in pose initialization situations for a known but uncooperative target spacecraft. The proposed algorithm finds convexity defect features from a target image and uses them as cues for matching feature points on the image to the points on the known target model. Based on this novel method for model matching, it estimates a pose by solving the PnP problem. Pose estimation simulations are carried out in three test scenarios, and each assesses the estimation accuracy and initialization performance by varying relative attitudes and distances. The simulation results show that the algorithm can estimate the poses of spacecraft models when a solar panel length and the number of solar panels are changed. Furthermore, a scenario considering the surface property of the spacecraft emphasizes that robust feature detection is essential for accurate pose estimation. This algorithm can be used for proximity operations with a known but uncooperative target spacecraft. Specifically, one of the main applications is relative navigation for on-orbit servicing.

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38

Maiyer, F. F., M. G. Tastanov, A. A. Utemissova, and S. A. Kozlovskiy. "EXACT ESTIMATES AND RADII OF CONVEXITY OF SOME CLASSES OF ANALYTIC FUNCTIONS." Bulletin of the South Ural State University series "Mathematics. Mechanics. Physics" 14, no. 1 (2022): 42–49. http://dx.doi.org/10.14529/mmph220105.

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The study of the geometric properties of analytic functions is one of the classical problems of the theory of functions of a complex variable and has been of steady interest to many mathematicians for more than half a century now. At the same time, a separate area is the building of sufficient conditions of one-leaf analytic functions, including finding the conditions for simple geometric properties of analytic functions (convex or star-shaped, almost starshaped, etc.). The solution of these problems in many cases is associated with finding estimates in different classes of analytical functions, which in itself is also a relevant problem. This article is devoted to finding exact estimates of analytic functions and their derivatives in fairly broad classes of functions, which are distinguished in the form of some restrictions on the domains obtained from the domains of values of these functions by circular symmetrization or symmetrization with respect to a straight line. Based on these results, the exact radii of convexity in some classes of functions are found.

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39

Ahmad, Khurshid, Muhammad Adil Khan, Shahid Khan, Amjad Ali, and Yu-Ming Chu. "New estimates for generalized Shannon and Zipf-Mandelbrot entropies via convexity results." Results in Physics 18 (September 2020): 103305. http://dx.doi.org/10.1016/j.rinp.2020.103305.

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40

Dolbeault, Jean, and Régis Monneau. "Convexity estimates for nonlinear elliptic equations and application to free boundary problems." Comptes Rendus de l'Académie des Sciences - Series I - Mathematics 331, no. 10 (November 2000): 771–76. http://dx.doi.org/10.1016/s0764-4442(00)01732-8.

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41

Dolbeault, Jean, and Régis Monneau. "Convexity estimates for nonlinear elliptic equations and application to free boundary problems." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 19, no. 6 (November 2002): 903–26. http://dx.doi.org/10.1016/s0294-1449(02)00106-3.

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42

Huisken, Gerhard, and Carlo Sinestrari. "Convexity estimates for mean curvature flow and singularities of mean convex surfaces." Acta Mathematica 183, no. 1 (1999): 45–70. http://dx.doi.org/10.1007/bf02392946.

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43

Zatitskiy, P. B., P. Ivanisvili, and D. M. Stolyarov. "Bellman VS. Beurling: sharp estimates of uniform convexity for $L^p$ spaces." St. Petersburg Mathematical Journal 27, no. 2 (January 29, 2016): 333–43. http://dx.doi.org/10.1090/spmj/1390.

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44

Ancona, Fabio, Olivier Glass, and Khai T. Nguyen. "On Kolmogorov Entropy Compactness Estimates for Scalar Conservation Laws Without Uniform Convexity." SIAM Journal on Mathematical Analysis 51, no. 4 (January 2019): 3020–51. http://dx.doi.org/10.1137/18m1198090.

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45

Lee, Mikyoung. "Weighted Orlicz regularity estimates for fully nonlinear elliptic equations with asymptotic convexity." Communications in Contemporary Mathematics 21, no. 04 (May 31, 2019): 1850024. http://dx.doi.org/10.1142/s0219199718500244.

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We prove interior Hessian estimates in the setting of weighted Orlicz spaces for viscosity solutions of fully nonlinear, uniformly elliptic equations [Formula: see text] under asymptotic assumptions on the nonlinear operator [Formula: see text] The results are further extended to fully nonlinear, asymptotically elliptic equations.

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46

Rasheed, Akhter, Saqib Hussain, Muhammad Asad Zaighum, and Maslina Darus. "Class of Analytic Function Related with Uniformly Convex and Janowski’s Functions." Journal of Function Spaces 2018 (October 16, 2018): 1–6. http://dx.doi.org/10.1155/2018/4679857.

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In this paper, we introduce a new subclass of analytic functions in open unit disc. We obtain coefficient estimates, extreme points, and distortion theorem. We also derived the radii of close-to-convexity and starlikeness for this class.

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47

Ullah, Hidayat, Muhammad Adil Khan, Tareq Saeed, and Zaid Mohammed Mohammed Mahdi Sayed. "Some Improvements of Jensen’s Inequality via 4-Convexity and Applications." Journal of Function Spaces 2022 (January 17, 2022): 1–9. http://dx.doi.org/10.1155/2022/2157375.

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The intention of this note is to investigate some new important estimates for the Jensen gap while utilizing a 4-convex function. We use the Jensen inequality and definition of convex function in order to achieve the required estimates for the Jensen gap. We acquire new improvements of the Hölder and Hermite–Hadamard inequalities with the help of the main results. We discuss some interesting relations for quasi-arithmetic and power means as consequences of main results. At last, we give the applications of our main inequalities in the information theory. The approach and techniques used in the present note may simulate more research in this field.

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48

Afzal, Waqar, and Thongchai Botmart. "Some novel estimates of Jensen and Hermite-Hadamard inequalities for h-Godunova-Levin stochastic processes." AIMS Mathematics 8, no. 3 (2023): 7277–91. http://dx.doi.org/10.3934/math.2023366.

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<abstract><p>It is undeniable that convex and non-convex functions play an important role in optimization. As a result of its behavior, convexity also plays a significant role in discussing inequalities. It is clear that convexity and stochastic processes are intertwined. The stochastic process is a mathematical model that describes how systems or phenomena fluctuate randomly. Probability theory generally says that the convex function applied to the expected value of a random variable is bounded above by the expected value of the random variable's convex function. Furthermore, the deep connection between convex inequalities and stochastic processes offers a whole new perspective on the study of inequality. Although Godunova-Levin functions are well known in convex theory, their properties enable us to determine inequality terms with greater accuracy than those obtained from convex functions. In this paper, we established a more refined form of Hermite-Hadamard and Jensen type inequalities for generalized interval-valued h-Godunova-Levin stochastic processes. In addition, we provide some examples to demonstrate the validity of our main findings.</p></abstract>

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49

Alessa, Nazek, B. Venkateswarlu, P. Thirupathi Reddy, K. Loganathan, and K. Tamilvanan. "A New Subclass of Analytic Functions Related to Mittag-Leffler Type Poisson Distribution Series." Journal of Function Spaces 2021 (February 1, 2021): 1–7. http://dx.doi.org/10.1155/2021/6618163.

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The object of this work is to an innovation of a class k − U ~ S T s ℏ , υ , τ , ι , ς in Y with negative coefficients, further determining coefficient estimates, neighborhoods, partial sums, convexity, and compactness of this specified class.

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50

Trudinger, Neil, and Feida Jiang. "Neumann problem for Monge-Ampere type equations revisited." New Zealand Journal of Mathematics 52 (October 24, 2021): 671–89. http://dx.doi.org/10.53733/176.

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This paper concerns a priori second order derivative estimates of solutions of the Neumann problem for the Monge-Amp\`ere type equations in bounded domains in n dimensional Euclidean space. We first establish a double normal second order derivative estimate on the boundary under an appropriate notion of domain convexity. Then, assuming a barrier condition for the linearized operator, we provide a complete proof of the global second derivative estimate for elliptic solutions, as previously studied in our earlier work. We also consider extensions to the degenerate elliptic case, in both the regular and strictly regular matrix cases.

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