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Strongly Log-Convex Functions Information Sciences Letters Volume 10 Issue 1 Jan. 2021 Article 5 2021 Strongly log-Convex Functions Muhammad Aslam Noor Department of Mathematics, COMSATS University Islamabad, Park Road, Islamabad, Pakistan, [email protected] Khalida Inayat Noor Department of Mathematics, COMSATS University Islamabad, Park Road, Islamabad, Pakistan, [email protected] Follow this and additional works at: https://digitalcommons.aaru.edu.jo/isl Recommended Citation Aslam Noor, Muhammad and Inayat Noor, Khalida (2021) "Strongly log-Convex Functions," Information Sciences Letters: Vol. 10 : Iss. 1 , Article 5. Available at: https://digitalcommons.aaru.edu.jo/isl/vol10/iss1/5 This Article is brought to you for free and open access by Arab Journals Platform. It has been accepted for inclusion in Information Sciences Letters by an authorized editor. The journal is hosted on Digital Commons, an Elsevier platform. For more information, please contact [email protected], [email protected], [email protected]. Aslam Noor and Inayat Noor: Strongly log-Convex Functions Inf. Sci. Lett. 10, No. 1, 33-38 (2021) 33 Information Sciences Letters An International Journal http://dx.doi.org/10.18576/isl/100105 Strongly log-Convex Functions Muhammad Aslam Noor∗ and Khalida Inayat Noor Department of Mathematics, COMSATS University Islamabad, Park Road, Islamabad, Pakistan Received: 2 Oct. 2020, Revised: 2 Nov.2020, Accepted: 12 Nov. 2020 Published online: 1 Jan. 2021 Abstract: Some new concepts of the strongly log convex functions are considered in this paper. Properties of the strongly convex functions are investigated under suitable conditions. The minimum of the differentiable strongly log-convex functions is characterized by variational inequality, which is itself an interesting problem. Some important special cases are discussed. It is proved that the parallelogram laws for inner product spaces can be obtained as applications of strongly log-affine functions as a novel application. Results obtained in this paper can be viewed as refinement and improvement of previously known results. Keywords: Convex functions, generalized convexity, log-convex functions, parallelogram laws, variational inequalities. 1 Introduction convex functions, see [13–18, 20–22]. Motivated and inspired by the ongoing research in generalized convexity, Convexity theory is a branch of mathematical sciences we consider strongly log-convex function, which is the with a wide range of applications in industry, physical, main motivation of this paper. Several new concepts of monotonicity are introduced. We establish the social, regional and engineering sciences. In recent years, several new generalizations of convex functions have relationship between these classes and derive some new been introduced using novel and innovative ideas to tackle results under some mild conditions. We have shown that difficult problems, which arise in various fields of pure new log-parallelogram laws can be obtained from strongly log-affine functions, which can be used to and applied sciences. Exponentially convex(concave) functions can be considered as a significant extension of characterize the inner product spaces. This fact can be the convex functions and have important applications in viewed itself an elegant and interesting applications of the strongly log-affine functions. As special cases, one can information theory, big data analysis, machine learning and statistic, see, for example, [1–8] and the references obtain various new and refined versions of known results. therein. Alirazaie and Mathur [2], and Noor et al [7–9] It is expected that the ideas and techniques of this paper have derived several results for these exponentially may stimulate further research in this field. convex functions. Related to the exponentially convex functions, we have log-convex functions. It is known that log-convex functions are convex functions, but the 2 Formulations and basic facts converse is not true. For example. the function ex is a log-convex function, but not convex. Hypergeometric Let K be a nonempty closed set in a real Hilbert space H. functions including Gamma and Beta functions are We denote by h·,·i and k · k be the inner productand norm, log-convex functions, which have important applications respectively. in several branches of pure and applied sciences. A different class of exponentially convex functions was Definition 1. [15, 16, 18] A set K in H is said to be a convex set, investigated by Awan et al. [10]. Noor et al [11] have if considered another equivalent formulation of log-convex u+t(v − u) ∈ K, ∀u,v ∈ K,t ∈ [0,1]. functions to discuss several new hidden aspects. We now introduce some new classes of strongly log-convex Strongly convex functions were introduced and functions and strongly log-affine functions. studied by Polyak [12]. For the applications of strongly ∗ Corresponding author e-mail: [email protected] c 2021 NSP Natural Sciences Publishing Cor. Published by Arab Journals Platform, 2021 1 Information Sciences Letters, Vol. 10 [2021], Iss. 1, Art. 5 34 M. A. Noor, K. I. Noor: Strongly log-convex functions Definition 2. A function F on the convex set K is said to be From the above definitions, we have strongly log- convex, if there exists a constant µ > 0, such that logF(u+t(v − u)) logF(u+t(v − u)) ≤ (1−t)logF(u) +t logF(v) ≤ (logF(u))1−t (logF(v))t − µt(1−t)kv − uk2 −µt(1−t)kv − u)k2 ,∀u,v ∈ K,t ∈ [0,1]. ≤ (1−t)logF(u) +t logF(v) − µt(1−t)kv − uk2 ≤ max{logF(u),logF(v)}− µt(1−t)kv − uk2 . We would like to mention that, if µ = 0, then Definition 2 reduces to This shows that every strongly log-convex function is a strongly log-convex function and every strongly log-convex function is a Definition 3. [11] A function F on the convex set K is said to strongly log-quasiconvex function. However, the converse is not be log- convex, if true. logF(u+t(v − u)) ≤ (1−t)log F(u) +t logF(v), ∀u,v ∈ K,t ∈ [0,1]. Definition 8. An operator T : K → H is said to be: For the properties of log-convex functions, see Noor et al. [11]. 1.strongly monotone, iff, there exists a constant α > 0 such that A function F is said to strongly log-concave, if and only if, 2 −F is strongly log-convex function. hTu− T v,u− vi ≥ αkv − uk ,∀u,v ∈ K. 1 If t = 2 , then 2.strongly pseudomonotone, iff, there exists a constant ν > 0 u+ v logF(u) + logF(v) such that logF ≤ 2 2 hTu,v − ui + νkv − uk2 ≥ 0 1 −µ kv − uk2,∀u,v ∈ K. ⇒ 4 hTv,v − ui ≥ 0,∀u,v ∈ K. The function F is said to be strongly J − log-convex function. 3.strongly relaxed pseudomonotone, iff, there exists a constant µ > 0 such that Note that if a functions is both strongly log-convex and hTu,v − ui ≥ 0 strongly log-concave, then it is strongly log-affine function. Consequently, we have new concepts. ⇒ −hTv,u− vi + µkv − uk2 ≥ 0,∀u,v ∈ K. Definition 4. A function F is said to be a strongly log-affine Definition 9. A differentiable function F on the convex set K is function, if there exists a constant µ > 0, such that said to be strongly log-pseudoconvex function, if and only if, if there exists a constant µ > 0 such that logF(u+t(v − u)) = (1−t)logF(u) +t logF(v) 2 F′(u) −µt(1−t)kv − uk ,∀u,v ∈ K,t ∈ [0,1]. h ,v − ui + µkv − uk2 ≥ 0 ⇒ F(v) ≥ F(u),∀u,v ∈ K. F(u) 1 If t = 2 , then Definition 4 reduces to: 3 Main results Definition 5.A function F is called strongly quadratic log-equation, if there exists a constant µ > 0, such that In this section, we consider some basic properties of strongly log- u+ v logF(u) + logF(v) 1 convex functions. logF = − µ kv − uk2, 2 2 4 Theorem 1. Let F be a differentiable function on the convex ∀u,v ∈ K,t ∈ [0,1]. set K. Then the function F is strongly log-convex function, if and only if, Definition 6. A function F on the convex set K is said to be strongly log-quasiconvex function, if there exists a constant µ > 0 F′(u) F(v) − F(u) ≥ h ,v − ui + µkv − uk2 ,∀v,u ∈ K. (1) such that F(u) logF(u+t(v − u) ≤ max{logF(u),logF(v)}− µt(1−t)kv − uk2 , Proof. Let F be a strongly log-convex function on the convex ∀u,v ∈ K,t ∈ [0,1]. set K. Then F(u+t(v − u)) ≤ (1−t)F(u) +tF(v) − µt(1−t)kv − uk2 , Definition 7. A function F on the convex set K is said to be strongly log-convex , if there exists a constant µ > 0 such that ∀u,v ∈ K, which can be written as logF(u+t(v − u)) ≤ (logF(u))1−t (logF(v))t 2 F(u+t(v − u) − F (u) −µt(1−t)kv − uk ,∀u,v ∈ K,t ∈ [0,1], F(v) − F(u) ≥ { }+ (1−t)µkv − uk2 , t where F(·) > 0. ∀u,v ∈ K. c 2021 NSP Natural Sciences Publishing Cor. https://digitalcommons.aaru.edu.jo/isl/vol10/iss1/5 2 Aslam Noor and Inayat Noor: Strongly log-Convex Functions Inf. Sci. Lett. 10, No. 1, 33-38 (2021) / www.naturalspublishing.com/Journals.asp 35 Taking the limit in the above inequality as t → 0,, we have Proof.Let F′ be a strongly monotone operator. Then, from (7), we have F′(u) F(v) − F(u) ≥ h ,v − u)i + µkv − uk2 ,∀u,v ∈ K. F′(v) F′(u) F(u) h ,u− vi ≥ h ,u− v)i + µkv − uk2 .
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