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Journal articles on the topic 'Concave and convex functions'

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Published: 10 December 2022
Last updated: 28 January 2023

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1

Borwein, David, and Julien Grivaux. "Convex and Concave Functions: 11009." American Mathematical Monthly 112, no. 1 (January 1, 2005): 92. http://dx.doi.org/10.2307/30037402.

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2

Meyer, M., G. Mokobodzki, and M. Rogalski. "Convex bodies and concave functions." Proceedings of the American Mathematical Society 123, no. 2 (February 1, 1995): 477. http://dx.doi.org/10.1090/s0002-9939-1995-1254848-2.

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3

Lynch, James D. "When Is a Renewal Process Convexly Parameterized in Its Mean Parameterization?" Probability in the Engineering and Informational Sciences 11, no. 1 (January 1997): 43–48. http://dx.doi.org/10.1017/s0269964800004666.

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The convex (concave) parameterization of a generalized renewal process is considered in this paper. It is shown that if the interrenewal times have log concave distributions or have log concave survival functions (i.e., an increasing failure rate distribution), then the renewal process is convexly (concavely) parameterized in its mean parameterization.
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4

Meyer, Clifford A., and Christodoulos A. Floudas. "Convex envelopes for edge-concave functions." Mathematical Programming 103, no. 2 (April 28, 2005): 207–24. http://dx.doi.org/10.1007/s10107-005-0580-9.

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5

Nemirovski, Arkadi. "On self-concordant convex–concave functions." Optimization Methods and Software 11, no. 1-4 (January 1999): 303–84. http://dx.doi.org/10.1080/10556789908805755.

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6

Moudafi, Abdellatif. "Well-behaved asymptotical convex-concave functions." Numerical Functional Analysis and Optimization 18, no. 9-10 (January 1997): 1013–21. http://dx.doi.org/10.1080/01630569708816806.

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7

Nikodem, Kazimierz. "Midpoint convex functions majorized by midpoint concave functions." Aequationes Mathematicae 31, no. 1 (December 1986): 322. http://dx.doi.org/10.1007/bf02188199.

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8

Nikodem, Kazimierz. "Midpoint convex functions majorized by midpoint concave functions." Aequationes Mathematicae 32, no. 1 (December 1987): 45–51. http://dx.doi.org/10.1007/bf02311298.

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9

Mateljevic, Miodrag, and Miloljub Albijanic. "Lagrange’s theorem, convex functions and Gauss map." Filomat 31, no. 2 (2017): 321–34. http://dx.doi.org/10.2298/fil1702321m.

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As one of the main results we prove that if f has Lagrange unique property then f is strictly convex or concave (we do not assume continuity of the derivative), Theorem 2.1. We give two different proofs of Theorem 2.1 (one mainly using Lagrange theorem and the other using Darboux theorem). In addition, we give a few characterizations of strictly convex curves, in Theorem 3.5. As an application of it, we give characterization of strictly convex planar curves, which have only tangents at every point, by injective of the Gauss map. Also without the differentiability hypothesis we get the characterization of strictly convex or concave functions by two points property, Theorem 4.2.
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10

SEGAL, ALEXANDER, and BOAZ A. SLOMKA. "PROJECTIONS OF LOG-CONCAVE FUNCTIONS." Communications in Contemporary Mathematics 14, no. 05 (August 29, 2012): 1250036. http://dx.doi.org/10.1142/s0219199712500368.

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Recently, it has been proven in [V. Milman, A. Segal and B. Slomka, A characterization of duality through section/projection correspondence in the finite dimensional setting, J. Funct. Anal. 261(11) (2011) 3366–3389] that the well-known duality mapping on the class of closed convex sets in ℝn containing the origin is the only operation, up to obvious linear modifications, that interchanges linear sections with projections. In this paper, we extend this result to the class of geometric log-concave functions (attaining 1 at the origin). As the notions of polarity and the support function were recently uniquely extended to this class by Artstein-Avidan and Milman, a natural notion of projection arises. This notion of projection is justified by our result. As a consequence of our main result, we prove that, on the class of lower semi continuous non-negative convex functions attaining 0 at the origin, the polarity operation is the only operation interchanging addition with geometric inf-convolution and the support function is the only operation interchanging addition with inf-convolution.
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11

Stuber, Matthew D., Joseph K. Scott, and Paul I. Barton. "Convex and concave relaxations of implicit functions." Optimization Methods and Software 30, no. 3 (June 17, 2014): 424–60. http://dx.doi.org/10.1080/10556788.2014.924514.

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12

Yuille, A. L., and Anand Rangarajan. "The Concave-Convex Procedure." Neural Computation 15, no. 4 (April 1, 2003): 915–36. http://dx.doi.org/10.1162/08997660360581958.

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The concave-convex procedure (CCCP) is a way to construct discrete-time iterative dynamical systems that are guaranteed to decrease global optimization and energy functions monotonically. This procedure can be applied to almost any optimization problem, and many existing algorithms can be interpreted in terms of it. In particular, we prove that all expectation-maximization algorithms and classes of Legendre minimization and variational bounding algorithms can be reexpressed in terms of CCCP. We show that many existing neural network and mean-field theory algorithms are also examples of CCCP. The generalized iterative scaling algorithm and Sinkhorn's algorithm can also be expressed as CCCP by changing variables. CCCP can be used both as a new way to understand, and prove the convergence of, existing optimization algorithms and as a procedure for generating new algorithms.
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13

Rahman, Atiqe Ur, Muhammad Saeed, Khuram Ali Khan, and Rostin Matendo Mabela. "Set-Theoretic Inequalities Based on Convex Multi-Argument Approximate Functions via Set Inclusion." Journal of Function Spaces 2022 (March 9, 2022): 1–10. http://dx.doi.org/10.1155/2022/6998104.

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Hypersoft set is a novel area of study which is established as an extension of soft set to handle its limitations. It employs a new approximate mapping called multi-argument approximate function which considers the Cartesian product of attribute-valued disjoint sets as its domain and the power set of universe as its co-domain. The domain of this function is broader as compared to the domain of soft approximate function. It is capable of handling the scenario where sub-attribute-valued sets are considered more significant than taking merely single set of attributes. In this study, notions of set inclusion, convex (concave) sets, strongly convex (concave) sets, strictly convex (concave) sets, convex hull, and convex cone are conceptualized for the multi-argument approximate function. Based on these characterized notions, some set-theoretic inequalities are established with generalized properties and results. The set-theoretic version of classical Jensen’s type inequalities is also discussed with the help of proposed notions.
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14

Beer, Gerald. "Quasi-concave functions and convex convergence to infinity." Bulletin of the Australian Mathematical Society 60, no. 1 (August 1999): 81–94. http://dx.doi.org/10.1017/s0004972700033359.

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By a convex mode of convergence to infinity 〈Ck〉, we mean a sequence of nonempty closed convex subsets of a normed linear space X such that for each k, Ck+1 ⊆ int Ck and and a sequence 〈xn〉 is X is declared convergent to infinity with respect to 〈Ck〉 provided each Ck contains xn eventually. Positive convergence to infinity with respect to a pointed cone with nonempty interior as well as convergence to infinity in a fixed direction fit within this framework. In this paper we study the representation of convex modes of convergence to infinity by quasi-concave functions and associated remetrizations of the space.
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15

LIEB, ELLIOTT H., and GERT K. PEDERSEN. "CONVEX MULTIVARIABLE TRACE FUNCTIONS." Reviews in Mathematical Physics 14, no. 07n08 (July 2002): 631–48. http://dx.doi.org/10.1142/s0129055x02001260.

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For any densely defined, lower semi-continuous trace τ on a C*-algebra A with mutually commuting C*-subalgebras A1, A2, … An, and a convex function f of n variables, we give a short proof of the fact that the function (x1, x2, …, xn)→ τ (f (x1, x2, …, xn)) is convex on the space [Formula: see text]. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called ℓ-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison–Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values.
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16

Kotrys, Dawid. "On Strongly Wright-Convex Stochastic Processes." Tatra Mountains Mathematical Publications 66, no. 1 (June 1, 2016): 67–72. http://dx.doi.org/10.1515/tmmp-2016-0020.

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Abstract Some characterizations of strongly Wright-convex stochastic processes are presented. Furthermore, the stochastic version of a theorem on strongly J-convex functions majorized by strongly J-concave functions is given.
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17

Khajavirad, Aida, and Nikolaos V. Sahinidis. "Convex envelopes of products of convex and component-wise concave functions." Journal of Global Optimization 52, no. 3 (July 15, 2011): 391–409. http://dx.doi.org/10.1007/s10898-011-9747-5.

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18

Gao, Fugen, and Meng Li. "Clarkson inequalities related to convex and concave functions." Mathematica Slovaca 71, no. 5 (October 1, 2021): 1319–27. http://dx.doi.org/10.1515/ms-2021-0055.

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Abstract In this paper, we obtain some norm inequalities involving convex and concave functions, which are the generalizations of the classical Clarkson inequalities. Let A 1, …, A n be bounded linear operators on a complex separable Hilbert space H $\mathcal{H}$ and let α 1, …, α n be positive real numbers such that ∑ j = 1 n α j = 1 $\sum\limits^{n}_{j=1}\alpha_{j}=1$ . We show that for every unitarily invariant norm, If f is a non-negative function on [0, ∞) such that f(0) = 0 and g ( t ) = f ( t ) $g(t)=f(\sqrt{t})$ is convex, then | | | ∑ j = 1 n α j f ( | A j | ) | | | ≥ | | | ∑ j , k ∈ S ℓ ( f ( α j α k 4 α ℓ ( 1 − α ℓ ) | A j + A k − 2 ∑ j = 1 n α j A j | ) + f ( α j α k ( 2 α ℓ − 1 ) 4 α ℓ ( 1 − α ℓ ) | A j − A k | ) ) + f ( | ∑ j = 1 n α j A j | ) | | | $$\begin{align*} \bigg|\bigg|\bigg|\sum\limits^{n}_{j=1}\alpha_{j}f(|A_{j}|)\bigg|\bigg|\bigg| &\geq\bigg|\bigg|\bigg|\sum\limits_{j,k\in S_{\ell}}\bigg(f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}}{4\alpha_{\ell}(1-\alpha_{\ell})}}\;\bigg|A_{j}+A_{k}-2\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\\ &\qquad+f\bigg(\sqrt{\frac{\alpha_{j}\alpha_{k}(2\alpha_{\ell}-1)}{4\alpha_{\ell}(1-\alpha_{\ell})}}|A_{j}-A_{k}|\bigg)\bigg)+f\bigg(\bigg|\sum\limits^{n}_{j=1}\alpha_{j}A_{j}\bigg|\bigg)\bigg|\bigg|\bigg| \end{align*}$$ for ℓ = 1, …, n. If f is a non-negative function on [0, ∞) such that g ( t ) = f ( t ) $g(t)=f(\sqrt{t})$ is concave, then the inverse inequality holds. Here, the symbol S ℓ = {1, …, n} ∖ {ℓ} for ℓ ∈ {1, …, n}. In addition, we provide some applications of the above inequalities.
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19

Thon, Dominique, and Lars Thorlund-Petersen. "Sums of increasing convex and increasing concave functions." Operations Research Letters 5, no. 6 (December 1986): 313–16. http://dx.doi.org/10.1016/0167-6377(86)90070-2.

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20

Chen, Chao-Ping, Feng Qi, Pietro Cerone, and Sever Silvestru Dragomir. "Monotonicity of sequences involving convex and concave functions." Mathematical Inequalities & Applications, no. 2 (2003): 229–39. http://dx.doi.org/10.7153/mia-06-22.

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21

Kamada, Yuichiro, and Fuhito Kojima. "Voter Preferences, Polarization, and Electoral Policies." American Economic Journal: Microeconomics 6, no. 4 (November 1, 2014): 203–36. http://dx.doi.org/10.1257/mic.6.4.203.

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In most variants of the Hotelling-Downs model of election, it is assumed that voters have concave utility functions. This assumption is arguably justified in issues such as economic policies, but convex utilities are perhaps more appropriate in others, such as moral or religious issues. In this paper, we analyze the implications of convex utility functions in a two-candidate probabilistic voting model with a polarized voter distribution. We show that the equilibrium policies diverge if and only if voters' utility function is sufficiently convex. If two or more issues are involved, policies converge in “concave issues” and diverge in “convex issues.” (JEL D72)
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22

Gorokhovik, V. V., and A. S. Tykoun. "The subdifferentiability of functions convex with respect to the set of Lipschitz concave functions." Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series 58, no. 1 (April 4, 2022): 7–20. http://dx.doi.org/10.29235/1561-2430-2022-58-1-7-20.

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A function defined on normed vector spaces X is called convex with respect to the set LĈ := LĈ (X,R ) ofLipschitz continuous classically concave functions (further, for brevity, LĈ -convex), if it is the upper envelope of some subset of functions from LĈ. A function f is LĈ -convex if and only if it is lower semicontinuous and bounded from below by a Lipschitz function. We introduce the notion of LĈ -subdifferentiability of a function at a point, i. e., subdifferentiability with respect to Lipschitz concave functions, which generalizes the notion of subdifferentiability of classically convex functions, and prove that for each LĈ -convex function the set of points at which it is LĈ -subdifferentiable is dense in its effective domain. The last result extends the well-known Brondsted – Rockafellar theorem on the existence of the subdifferential for classically convex lower semicontinuous functions to the more wide class of lower semicontinuous functions. Using elements of the subset LĈ θ ⊂ LĈ, which consists of Lipschitz continuous functions vanishing at the origin of X we introduce the notions of LĈ θ -subgradient and LĈ θ -subdifferential for a function at a point.The properties of LĈ -subdifferentials and their relations with the classical Fenchel – Rockafellar subdifferential are studied. Considering the set LČ := LČ (X,R ) of Lipschitz continuous classically convex functions as elementary ones we define the notions of LČ -concavity and LČ -superdifferentiability that are symmetric to the LĈ -convexity and LĈ -subdifferentiability of functions. We also derive criteria for global minimum and maximum points of nonsmooth functions formulated in terms of LĈ θ -subdifferentials and LČ θ -superdifferentials.
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23

Jakšetić, Julije. "One Concave-Convex Inequality and Its Consequences." Mathematics 9, no. 14 (July 12, 2021): 1639. http://dx.doi.org/10.3390/math9141639.

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Our starting point is an integral inequality that involves convex, concave and monotonically increasing functions. We provide some interpretations of the inequality, in terms of both probability and terms of linear functionals, from which we further generate completely monotone functions and means. The latter application is seen from the perspective of monotonicity and convexity.
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24

Latif, M. A., and S. S. Dragomir. "New inequalities of Hermite-Hadamard type for n-times differentiable convex and concave functions with applications." Filomat 30, no. 10 (2016): 2609–21. http://dx.doi.org/10.2298/fil1610609l.

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In this paper, a new identity for n-times differntiable functions is established and by using the obtained identity, some new inequalities Hermite-Hadamard type are obtained for functions whose nth derivatives in absolute value are convex and concave functions. From our results, several inequalities of Hermite-Hadamard type can be derived in terms of functions whose first and second derivatives in absolute value are convex and concave functions as special cases. Our results may provide refinements of some results already exist in literature. Applications to trapezoidal formula and special means of established results are given.
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25

Gorokhovik, V. V., and A. S. Tykoun. "Support points of lower semicontinuous functions with respect to the set of Lipschitz concave functions." Doklady of the National Academy of Sciences of Belarus 63, no. 6 (January 7, 2020): 647–53. http://dx.doi.org/10.29235/1561-8323-2019-63-6-647-653.

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For the functions defined on normed vector spaces, we introduce a new notion of the LC -convexity that generalizes the classical notion of convex functions. A function is called to be LC -convex if it can be represented as the upper envelope of some subset of Lipschitz concave functions. It is proved that the function is LC -convex if and only if it is lower semicontinuous and, in addition, it is bounded from below by a Lipschitz function. As a generalization of a global subdifferential of a classically convex function, we introduce the set of LC -minorants supported to a function at a given point and the set of LC -support points of a function that are then used to derive a criterion for global minimum points and a necessary condition for global maximum points of nonsmooth functions. An important result of the article is to prove that for a LC - convex function, the set of LC -support points is dense in its effective domain. This result extends the well-known Brondsted– Rockafellar theorem on the existence of the sub-differential for classically convex lower semicontinuous functions to a wider class of lower semicontinuous functions and goes back to the one of the most important results of the classical convex analysis – the Bishop–Phelps theorem on the density of support points in the boundary of a closed convex set.
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26

Chen, Fangwei, Jianbo Fang, Miao Luo, and Congli Yang. "On Mixed Quermassintegral for Log-Concave Functions." Journal of Function Spaces 2020 (November 17, 2020): 1–9. http://dx.doi.org/10.1155/2020/8811566.

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In this paper, the functional Quermassintegral of log-concave functions in ℝ n is discussed. We obtain the integral expression of the i th functional mixed Quermassintegral, which is similar to the integral expression of the i th mixed Quermassintegral of convex bodies.
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27

Wang, Hongxia. "Fubini Theorems for Capacities." International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 24, no. 06 (November 30, 2016): 901–16. http://dx.doi.org/10.1142/s0218488516500410.

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Capacity plays an important role in many areas. A capacity is usually studied under the assumption that it is concave (or convex). In this paper, we perform a further investigation on the Fubini Theorems for concave (or convex) capacities given by Ghirardato (1997) and Chateauneuf and Lefort (2008). We extend Fubini Theorems for capacities to a larger class of functions, which are both μ1-Choquet integrable and μ2-Choquet integrable, not to restrict only on slice-comonotonic functions.
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28

Chen, Feixiang. "Generalizations of Inequalities for Differentiable Co-Ordinated Convex Functions." Chinese Journal of Mathematics 2014 (March 19, 2014): 1–12. http://dx.doi.org/10.1155/2014/741291.

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29

Abbas Baloch, Imran, and Yu-Ming Chu. "Petrović-Type Inequalities for Harmonic h-convex Functions." Journal of Function Spaces 2020 (January 8, 2020): 1–7. http://dx.doi.org/10.1155/2020/3075390.

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In the article, we establish several Petrović-type inequalities for the harmonic h-convex (concave) function if h is a submultiplicative (super-multiplicative) function, provide some new majorizaton type inequalities for harmonic convex function, and prove the superadditivity, subadditivity, linearity, and monotonicity properties for the functionals derived from the Petrović type inequalities.
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30

Agarwal, Praveen, Mahir Kadakal, İmdat İşcan, and Yu-Ming Chu. "Better Approaches for n-Times Differentiable Convex Functions." Mathematics 8, no. 6 (June 10, 2020): 950. http://dx.doi.org/10.3390/math8060950.

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In this work, by using an integral identity together with the Hölder–İşcan inequality we establish several new inequalities for n-times differentiable convex and concave mappings. Furthermore, various applications for some special means as arithmetic, geometric, and logarithmic are given.
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31

Alomari, Mohammad W. "A Sharp Bound for the Čebyšev Functional of Convex or Concave Functions." Chinese Journal of Mathematics 2013 (September 23, 2013): 1–3. http://dx.doi.org/10.1155/2013/295146.

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32

Hue, Nguyen Ngoc, and Duong Quoc Huy. "Monotonicity of sequences involving generalized convexity function and sequences." Tamkang Journal of Mathematics 46, no. 2 (June 30, 2015): 121–27. http://dx.doi.org/10.5556/j.tkjm.46.2015.1626.

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In this paper, by using the theory of generalized convexity functions we introduce and prove monotonicity of sequences of the forms $$ \left\{\left(\prod\limits_{k=1}^nf\left({a_k\over a_n}\right)\right)^{1/n}\right\},\quad \left\{\left(\prod\limits_{k=1}^nf\left({\varphi(k)\over\varphi(n)}\right)\right)^{1/\varphi(n)}\right\}, $$ $$ \left\{{1\over n}\sum_{k=1}^nf\left({a_n\over a_k}\right)\right\}\quad\text{or}\quad \left\{{1\over\varphi(n)}\sum_{k=1}^nf\left({\varphi(n)\over\varphi(k)}\right)\right\}, $$ where $f$ belongs to the classes of $AG$-convex (concave), $HA$-convex (concave), or $HG$-convex (concave) functions defined on suitable intervals, $\{a_n\}$ is a given sequence and $\varphi$ is a given function that satisfy some preset conditions. As a consequence, we obtain some generalizations of Alzer type inequalities.
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33

Bekjan, Turdebek N., Kordan N. Ospanov, and Asilbek Zulkhazhav. "Choi-Davis-Jensen Inequalities in Semifinite von Neumann Algebras." Journal of Function Spaces 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/208923.

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34

Hong, Dug Hun, and Jae Duck Kim. "Hölder Type Inequalities for Sugeno Integrals under Usual Multiplication Operations." Advances in Fuzzy Systems 2019 (January 3, 2019): 1–10. http://dx.doi.org/10.1155/2019/5080723.

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The classical Hölder inequality shows an interesting upper bound for Lebesgue integral of the product of two functions. This paper proposes Hölder type inequalities and reverse Hölder type inequalities for Sugeno integrals under usual multiplication operations for nonincreasing concave or convex functions. One of the interesting results is that the inequality, (S)∫01f(x)pdμ1/p(S)∫01g(x)qdμ1/q≤p-q/p-p-q+1∨q-p/q-q-p+1(S)∫01f(x)g(x)dμ, where 1<p<∞,1/p+1/q=1 and μ is the Lebesgue measure on R, holds if f and g are nonincreasing and concave functions. As a special case, we consider Cauchy-Schwarz type inequalities for Sugeno integrals involving nonincreasing concave or convex functions. Some examples are provided to illustrate the validity of the proposed inequalities.
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35

Merhav, Neri. "Reversing Jensen’s Inequality for Information-Theoretic Analyses." Information 13, no. 1 (January 13, 2022): 39. http://dx.doi.org/10.3390/info13010039.

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In this work, we propose both an improvement and extensions of a reverse Jensen inequality due to Wunder et al. (2021). The new proposed inequalities are fairly tight and reasonably easy to use in a wide variety of situations, as demonstrated in several application examples that are relevant to information theory. Moreover, the main ideas behind the derivations turn out to be applicable to generate bounds to expectations of multivariate convex/concave functions, as well as functions that are not necessarily convex or concave.
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36

Azé, Dominique, Hedy Attouch, and Roger J. B. Wets. "Convergence of convex-concave saddle functions: applications to convex programming and mechanics." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 5, no. 6 (November 1988): 537–72. http://dx.doi.org/10.1016/s0294-1449(16)30335-3.

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37

Veremyev, Alexander, Peter Tsyurmasto, Stan Uryasev, and R. Tyrrell Rockafellar. "Calibrating probability distributions with convex-concave-convex functions: application to CDO pricing." Computational Management Science 11, no. 4 (July 10, 2013): 341–64. http://dx.doi.org/10.1007/s10287-013-0176-4.

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38

Fang, Niufa, and Jin Yang. "Functional Geominimal Surface Area and Its Related Affine Isoperimetric Inequality." Journal of Function Spaces 2020 (July 13, 2020): 1–8. http://dx.doi.org/10.1155/2020/3039598.

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The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.
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39

Han Wu Chen and K. Yanagi. "The convex-concave characteristics of Gaussian channel capacity functions." IEEE Transactions on Information Theory 52, no. 5 (May 2006): 2167–72. http://dx.doi.org/10.1109/tit.2006.872851.

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40

Sulaiman, WaadAllah T. "INTEGRAL INEQUALITY REGARDING γ-CONVEX AND γ-CONCAVE FUNCTIONS." Journal of the Korean Mathematical Society 47, no. 2 (March 1, 2010): 373–83. http://dx.doi.org/10.4134/jkms.2010.47.2.373.

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41

Boche, H., and M. Schubert. "Concave and Convex Interference Functions—General Characterizations and Applications." IEEE Transactions on Signal Processing 56, no. 10 (October 2008): 4951–65. http://dx.doi.org/10.1109/tsp.2008.928093.

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42

Alrimawi, Fadi, Omar Hirzallah, and Fuad Kittaneh. "Norm inequalities involving convex and concave functions of operators." Linear and Multilinear Algebra 67, no. 9 (May 7, 2018): 1757–72. http://dx.doi.org/10.1080/03081087.2018.1470601.

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43

Hussain, S., J. Pečarić, and I. Perić. "Jensen's Inequality for Convex-Concave Antisymmetric Functions and Applications." Journal of Inequalities and Applications 2008 (2008): 1–6. http://dx.doi.org/10.1155/2008/185089.

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44

BELBACHIR, HACENE, and MOURAD RAHMANI. "Bounds for the generalized Lupas functional." Creative Mathematics and Informatics 20, no. 1 (2011): 24–31. http://dx.doi.org/10.37193/cmi.2011.01.07.

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45

Ekinci, Alper, and Nazlıcan Eroğlu. "New generalizations for convex functions via conformable fractional integrals." Filomat 33, no. 14 (2019): 4525–34. http://dx.doi.org/10.2298/fil1914525e.

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The main objective of this study is to obtain conformable analogs of some inequalities and to examine some integral inequalities for functions whose modulus of first derivatives are convex and concave. We obtain generalizations including conformable fractional integrals by separating [a; b] interval to s equal subintervals.
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46

Fang, Niufa, and Zengle Zhang. "The Minimal Perimeter of a Log-Concave Function." Mathematics 8, no. 8 (August 14, 2020): 1365. http://dx.doi.org/10.3390/math8081365.

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Inspired by the equivalence between isoperimetric inequality and Sobolev inequality, we provide a new connection between geometry and analysis. We define the minimal perimeter of a log-concave function and establish a characteristic theorem of this extremal problem for log-concave functions analogous to convex bodies.
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47

Gal, Sorin Gheorghe. "Properties of the modulus of continuity for monotonous convex functions and applications." International Journal of Mathematics and Mathematical Sciences 18, no. 3 (1995): 443–46. http://dx.doi.org/10.1155/s016117129500055x.

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48

Simić, Slavko, and Bandar Bin-Mohsin. "Simpson’s Rule and Hermite–Hadamard Inequality for Non-Convex Functions." Mathematics 8, no. 8 (July 31, 2020): 1248. http://dx.doi.org/10.3390/math8081248.

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In this article we give a variant of the Hermite–Hadamard integral inequality for twice differentiable functions. It represents an improvement of this inequality in the case of convex/concave functions. Sharp two-sided inequalities for Simpson’s rule are also proven along with several extensions.
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49

Ohwa, Hiroki. "An 𝐿^{𝑝} shock admissibility condition for conservation laws." Quarterly of Applied Mathematics 80, no. 2 (February 1, 2022): 259–75. http://dx.doi.org/10.1090/qam/1610.

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We estimate the L p L^p ( p > 0 p>0 ) local distance between piecewise constant solutions to the Cauchy problem of conservation laws and propose a shock admissibility condition for having an L p L^p local contraction of such solutions. Moreover, as an application, we prove that there exist L p L^p locally contractive solutions on some set of initial functions, to the Cauchy problem of conservation laws with convex or concave flux functions. As a result, for conservation laws with convex or concave flux functions, we see that rarefaction waves have an L q L^q ( q ≥ 1 q\geq 1 ) local contraction and shock waves have an L r L^r ( 0 > r ≤ 1 0>r\leq 1 ) local contraction.
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50

Shanthikumar, J. George. "Stochastic majorization of random variables by proportional equilibrium rates." Advances in Applied Probability 19, no. 4 (December 1987): 854–72. http://dx.doi.org/10.2307/1427105.

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The equilibrium rate rY of a random variable Y with support on non-negative integers is defined by rY(0) = 0 and rY(n) = P[Y = n – 1]/P[Y – n], Let (j = 1, …, m; i = 1,2) be 2m independent random variables that have proportional equilibrium rates with (j = 1, …, m; i = 1, 2) as the constant of proportionality. When the equilibrium rate is increasing and concave [convex] it is shown that , …, ) majorizes implies , …, for all increasing Schur-convex [concave] functions whenever the expectations exist. In addition if , (i = 1, 2), then
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