Some Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functionsSome Fejér type inequalities for strongly (Mϕ,Mψ) convex functions
KHOA HOG TRUONG DAI HOC QUY NHON Một số bất đẳng thức kiểu Fejér cho hàm (4, A⁄4,) - lồi mạnh Nguyễn Ngọc Huề” Khoa Khoa hoc Tu nhiên va Cong nghé, Dai hoc Tay Nguyén, Dak Lắk, Việt Nam Ngàu nhận bài: 30/03/2022; Ngàu nhận đăng: 07/06/2029; Ngày xuất bản: 28/10/2022 TĨM TẮT Trong báo này, chúng tơi xem xét lớp hàm lỗi mạnh mở rộng liên quan đến cặp tựa trung bình số học, gọi hàm (,MI¿, M„)-lồi mạnh từ thiết lập số bất đẳng thức kiểu Fejér cho lớp hàm lồi mạnh Các bất đẳng thức mở rộng thực bất đẳng thức Hermite-Hadamard bất đẳng thức Fejér thiết lập gần hàm lồi mạnh số dạng mở rộng lớp hàm lồi mạnh Hơn nữa, bất đẳng thức đặc trưng cho lớp ham (M4, M,)-16i manh Từ khóa: Hàm (M„,.M1,)-lồi mạnh, tựa trưng bình số học, hàm lồi, bắt đẳng thúc Hermite-Hadamard, bất đẳng thúc Fjér * Tác giả liên hệ Email: nnhue@ttn.edu.vn https://doi.org/10.52111/qnjs.2022.16508 Tap chi Khoa hoc - Trường Đại học Quy Nhơn, 2022, 16(5), 87-95 | 87 BEETETTEON SCIENCE QUY NHON UNIVERSITY Some Fejér type inequalities for strongly (Mg, My) - convex Nguyen Department of Technology and Science, Ngoc functions Hue* Tay Nguyen University, Dak Lak, Vietnam Received: 30/08/2022; Accepted: 07/06/2022; Published: 28/10/2022 ABSTRACT In this paper, we propose and study a class of generalized convex functions, which are defined according to a pair of quasi-arithmetic means and called (My, M,)-convex functions, and establish various Fejér type inequalities for such a function class These inequalities not merely provide a natural and intrinsic characterization of the (My, My,)-convex functions, but actually offer a generalization and refinement of some Hermite-Hadamard and Fejér type inequalities obtained in earlier studies for different kinds of strong convexity Keywords: Strongly (My, M,)-convex functions, quasi-arithmetic mean, convexity, Hermite-Hadamard inequality, Fejér inequality INTRODUCTION Let In the field of mathematical inequalities, the wellknown Hermite-Hadamard inequality for convex functions was first discovered by Hermite! in 1883 and independently discovered 10 years later by Hadamard.” This inequality says that if f : [a,b] > R is a convex function then /(°)< bền J_ 76 < f8>8, A weighted generalization of Hermite-Hadamard inequality was developed by Fejér:? If f : [a,b] > R is a convex function, g : [ø, b] — [0, 00) is an integrable function with fe g(z)dx > and it is symmetric to at! i.e g(x) =g(a+b—2z) for all x € [a,}], then /{ a+b) )* fo f@glede — fla) + 70) Íÿ g(œ)dz (2) Since then, the inequalities (1) and (2) have been generalized, extended and improved in various ways and found interesting applications to convex analysis, optimization theory and nonlinear analysis One of such ways is to establish new inequalities for various generalized convex functions (see e.g.,*’) Among them, an important subclass of convex functions in the optimization theory is strongly convex functions This class was developed by Polyak® in 1966 for dealing with some related issues arisen from optimization theory c be a positive number A function f : [a,b] — R is called strongly convex with modulus c if f(ta+(1—t)y) < tf(a)+(1—-t) f(y) —et(1—t)(@-y)? for all x,y € [a,b] One says that f is strongly midconvex with modulus c if (22) < feet x (x — 9)? (3) for all x,y € [a, 5] In 2010, Merentes and Nikodem® established a generalized version of Hermite-Hadamard inequality for strongly convex functions as follows: Let f : [a,b] > R be a strongly convex function with modulus c Then, the following inequality holds (et) 1a (ð— ø} fla) +1 b) _¢ gio 4) (4) In 2012, Azocar!2 et al proposed a Fejér type inequality for strongly convex functions: Let f [a,b] + R bea strongly convex function with modulus c and g : [a, b] —> [0, 00) be an integrable function *Corresponding author Email: nnhue@ttn.edu.vn https://doi.org/10.52111/qnjs.2022.16508 88 | Quy Nhon University Journal of Science, 2022, 16(5), 87-95 : BEETETTEON SCIENCE QUY NHON UNIVERSITY with ƒ ø(z)dz = and symmetric to *£ Then, Definition Let c be a positive number A function f : I > J is called strongly (My, M,)-convex with modulus c if f (=) +e [ 2sœ&- (3 < [ Te)g6)4 (5 < f+ Ff) _, e+e - [eae functions ‘1-12 Motivated by the achievements, we continue the research direction Our contributions in this deeply are that we first investigate the class of generalized strongly convex functions regarding to a pair of quasi-arithmetic means and then derive some new Fejér-type inequalities The derived inequalities are not only characterizations for generalized strongly convex continuous functions, but they also generalize inequalities that were recently derived in the papers !!:12 Section 2, we will introduce (M4, My,)-convex functions, strongly (Mg,M,)-convex functions with and refer related particular cases The main results of this paper will be presented in Section Finally, the paper closes with the conclu- sions in Section STRONGLY (Mg,M,) - CONVEX FUNCTIONS Let J bers Let continuous a pair of and J be the intervals of real numó : J > R and w J — R be and strictly monotone functions Using quasi-arithmetic means Mg and My, with M4(a,b;a) = $ | (ad(a) + (1 — a) ¢(b)) Aumann!* proposed the concept of (M4, M,)-convex functions that is stated as follows Definition '° A function f : I > J is said to be (M¢, My)-convex if f(Mg(a,b;a)) < My(F(@), fa) — (6) for all a,b € J and a € (0, 1] In the case that f fulfills the inequality (6) with @(z) = 2, f is called My-convex If f satisfies the inequality (6) with ¢(x) = x and v(x) = z, then the (My, M,)-convexity of f reduces to the usual convexity in the literature of convex analysis For a pair of quasi-arithmetic (7) is reversed, we call that f is strongly (Mg,M,)concave with modulus c Note that if ~ is increasing then f : I > J is strongly (My, M,)-convex with modulus c if and only if fod! is strongly convex function with modulus c on ¢(J) If ~ is decreasing then f : I > J is strongly (My, M,)-convex with modulus c wo fod iff is strongly concave with modulus e on 9) We say that a function ƒ is strongly {„-convex with modulus c if f satisfies the inequality (7) for $() = x For particular forms of ¢ and w, we obtain the following concepts: * strongly convex functions if we take ¢(z) = x The rest of this paper is organized as follows In positive modulus —eo(1— a)(6(a) — #(8))?) for all a,b € I and a € [0,1] If the inequality (7) In recent years, some generalizations of inequality (5) have been established for strongly logconvex functions and strongly harmonic convex paper ƒ(Mu(a,b;a)) < ~ (ae ƒ(a) + (1— a)# e ƒ(0) means My and My, we define a class of generalized strongly convex functions as follows and W(x) =a: f(aa+t (1 —a)b) < af(a) + (1— a) f(b) — ca(1 — a)(a — 6)? for alla,b € I anda é * Strongly log-convex (0,1) functions’4+ o(z) = and (z) = laz: if we take In ƒ(œa + (1 — œ)b) < aln f(a) + (1 — a) In f(b) — ca(1 — a)(a — b)? for alla,b€ I anda é * Strongly exponentially [0,1] convex functions? we take @(%) = x and W(x) = e”: if ef (aat(1—a)b) < aef + (1 — a)ef — ca(1 — a)(a — b)? for alla,b€ I anda € (0,1) * Strongly harmonic convex functions'! take @(œ) = 1/ax and (x) = 2: if we j ester) < af(a) + (1 — a) f(b) — ca(1 — a) (2) Jor aÏl a,b€ T ơnd œ € |0, 1J * Strongly harmonic log-convex functions’! we take (x) = 1/ax and (z) = Ìnz: if f (Gsra=an) < /(9*ƒ0)*~® = sag= a) (* ty for all a,b € I anda € [0,1] https://doi.org/10.52111/qnjs.2022.16508 Quy Nhon University Journal of Science, 2022, 16(5), 87-95 | 89 BEETETTEON SCIENCE QUY NHON UNIVERSITY * Strongly p-convex functions if $() = +P and W(x) =a: f (laa? + (1 — @)b?]/?) < af(a) + (1— a) f(b) — ca(l — a) (a? — bP)? for alla,b € I anda € and G(t) =tF(1) + (1 —t)F(0) Then, (1) F and G are My-convex, F (0) = G(0) * Strongly geometrically convex functions if we take $(x) =Inz and (z) =lnz: F(t) J is strongly midconvex with modulus c if and only if g(x) = f(x)—cx? and Jensen’s inequality !”, one can verify that if f is continuous on J and strongly midconvex with modulus c then f is a strongly convex function with modulus c on I and (8) [SoYo F(tur (dt Lm) ja Jo twi (t)dt Pr(s) Then, F o Bi, I, (0, 1] and satisfy tì (ái and Go fy are increasing on lim, Fof(s) = lim Z,(s) = lim, Gofi(s) = 9(0), s—>0T Fo Bi(s) J is strongly (M4,M,)-convex with modulus c if and only if g(x) := Wo fod -*(x) — ca? is convex on (1) LÌ FEJER TYPE INEQUALITIES FOR STRONGLY (M4, M,y)-CONVEX FUNCTIONS Throughout this paper, we assume that ƒ : Ï > J is a strongly (My,M,)-convex function with modulus 0); a,b b; a € c (c > € I, a < (0,1); W 1, W2 : [0,1] — [0, 00) are integrable functions and satisfy the condition f} w1(¢)dt > for all s € (0, 1] and f wo(t)dt > for all s € [0,1) Denote L(t) = Mo(a, Mg(a, b; a); t) and _ Jtwo(t)at P= F Ba(s) < T2(s) < Gofe(s), lim s1- Z#ofØa(s) R(t) = Mo(b, Mola, 6; a); t) for t € [0, 1] + (1—a)(bo foR(t)— [bo R(O?)) im T(s) = im GoBo(s) = G(1) In order to prove Theorem 5, we need to intro- duce the following auxiliary result ° Let P : [0,1] > R be increasing and continuous (1) For s € (0,1), define Theorem Let F,G: [0,1] > R be two functions defined as F(t) = (awe foci ~ obo £(t))?) — s € (0,1), (10) Moreover, if wy = we then T,(1) = 7a(0) Lemma and a(iat Then, F o Bz, Tz and Go By are increasing on |0, 1) and satisfy Py (s) = So P(t)wi(t)at fo wi(t)dt Then, P, is increasing on (0,1] and lim P,(s) = P(0) < P,(s) < P(s), s € (0,1] s—Ot https://doi.org/10.52111/qnjs.2022.16508 90 | Quy Nhon University Journal of Science, 2022, 16(5), 87-95 BEETETTEON SCIENCE QUY NHON UNIVERSITY (2) Similarly, for s € [0,1), define py) and Pe ? vo foo (BW) ~ BI) fiw _ we < ¿(0e ƒ()— c(ð(ð))?) + (1—t)(bo f(Mog(a,b;, a)) Then, Pz is increasing on (0,1) and P(s) < Pa(s) < P(1) = lim Pa(s), » € (0,1) — e(ad(a) + (1 — a) ¢(6))?) Thus, We are now in a position to provide the proof of Theorem poF(t) 1— lim fa(s)=1> f2(s)>s, 2a(s) (4) For alla,b € I anda for all s € (0,1] and Ƒ g2 ° R(t)dt > for all s € [0,1) and satisfy (15), (16) Then, (i) For all s € (0,1], we have yo ( ° /(Mu(a,b;ø)) — e(að(a) + (1- a)6(0)?) ƒ o £0)dt