The main results of this paper give a connection between strong Jensen convexity and strong convexity

type inequalities. We are also looking for the optimal Takagi type function of strong convexity.

Finally a connection will be proved between the Jensen error term and an useful error function

Házy, Attila

Makó, Judit

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On strongly convex functionsJudit Makó and Attila Házy1Institute of Mathematics, University of Miskolc3515 MISKOLC-EGYETEMVÁROS, Hungary{matjudit, matha}@uni-miskolc.hu1. IntroductionThroughout this paper R, R+, N and Z denote the sets of real, nonnegative real, natural and integer numbersrespectively.Let X be a normed space and D ⊆ X be a nonempty conex subset of X. Denote by D∗ the set {‖x−y‖, x, y ∈D}. Let α : D∗ → R+ be a nonnegative error function. We say that a function f : D → R is strongly α-Jensenconvex, if, for all x, y ∈ D,(1) f(x+ y2)≤ f(x) + f(y)2− α(‖x− y‖).Observe that if α ≡ 0, we can get the classical definition of convexity. When α(u) = cu2, we can get a kindof notion of strong convexity introduced by Polyak in [24] and examined by Azócar, Giménez, Nikodem andSánchez (in [1]), Merentes and Nikodem, (in [16]) and Nikodem and Páles [22]. If α(u) = εup, then f is calledstrongly (ε, p)-Jensen convex function. In Section 2, we are looking connection between strong α-convexity andstrong convexity type inequalities. Then, we are looking for the optimal error function. In Section 3, we willestablish the connections between strong α-convexity and strong α-Jensen convexity, moreover the connectionsbetween strong convexity and HermiteHadamard type inequalities will be shown. In what follows we recallsome Bernstein-Doetsch type theorem for approximately convex functions. A function f : D → R is said to beapproximately α-Jensen convex, if, for all x, y ∈ D,(2) f(x+ y2)≤ f(x) + f(y)2+ α(‖x− y‖).The proofs of these theorem are similar to our main theorems' proofs.Let introduce the Takagi type functions Tα : R×D+ → R+ and Sα : R×D+ → R+ by(3) Tα(t, u) :=∞∑n=012nα(dZ(2nt)u) ((t, u) ∈ R×D+)and(4) Sα(t, u) :=∞∑n=0α( u2n)dZ(2nt)((t, u) ∈ R×D+).Note that the first series converges uniformly if α is bounded, on the other hand, for the uniform convergenceof the second series, it is sufficient if∑∞n=n0α(2−n)<∞ for some n0 ∈ N.The importance of the function Tα introduced above is enlightened by the following result which can beconsidered as a generalization of the celebrated Bernstein-Doetsch theorem [2].Theorem 1. (Makó-Páles [15], TaborTabor [26])Let f : D → R be locally upper bounded on D and let α : D+ → R+. Then f is α-Jensen convex on D if and onlyif(5) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y) + Tα(t, ‖x− y‖)for all x, y ∈ D and t ∈ [0, 1].The other Takagi type function Sα was introduced by Jacek Tabor and Józef Tabor. Its role and importancein the theory of approximate convexity is shown by the next theorem.1This research has been supported:1. by the Hungarian Scientific Research Fund (OTKA) Grant NK-81402 and and by the European Union and the State ofHungary, co-financed by the European Social Fund in the framework of TÁMOP-4.2.4.A/ 2-11/1-2012-0001 `National ExcellenceProgram';2. by the János Bolyai Research Scholarship of the Hungarian Academy of Sciences.2000 Mathematics Subject Classification: 39B22, 39B12keywords: strong convexity, strong Jensen convexity, Takagi-type functions1Theorem 2. (Tabor-Tabor [26])Let f : D → R be upper semicontinuous on D and let α : D+ → R+ be nondecreasing such that∑∞n=n0α(2−n) <∞ for some n0 ∈ N. Then f is α-Jensen convex on D if and only if(6) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y) + Sα(t, ‖x− y‖)for all x, y ∈ D and t ∈ [0, 1].Let ε, q ≥ 0 be arbitrary constants. When α(u) := εuq, (u ∈ D+), the two corollaries below (see [8] and [26])are immediately consequences of the previous theorems.For q ≥ 0, define the Takagi type functions Sq and Tq by(7) Tq(t) :=∞∑n=0(dZ(2nt))q2n, Sq(t) :=∞∑n=0dZ(2nt)2nq(t ∈ R).They generalize the classical Takagi functionT (t) :=∞∑n=0dist(2nt,Z)2n(t ∈ R)in two ways, because T1 = S1 = 2T holds obviously. This function was introduced by Takagi in [29] and it is awell-known example of a continuous but nowhere differentiable real function.It is less trivial, but it can be proved that T2(t) = S2(t) = 4t(1 − t) for t ∈ [0, 1]. The following picturesdemonstrate the comparison between Tq and Sq for q = 0.5 and q = 1.5, respectively.1,60,81,220,41,50-1 0 1x0,5-0,50,40,20,320,100 1x0,5-0,5 1,5-1Corollary 3. (Házy [4])Let f : D → R be locally upper bounded on D and ε, q ≥ 0. Then f is (ε, q)-Jensen convex on D, if and only if(8) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y) + εTq(t)‖x− y‖qfor all x, y ∈ D and t ∈ [0, 1].Corollary 4. (Tabor-Tabor [26])Let f : D → R be upper semicontinuous on D and ε, q ≥ 0. Then f is (ε, q)-Jensen convex on D if and only if(9) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y) + εSq(t)‖x− y‖qfor all x, y ∈ D and t ∈ [0, 1].In [3], Boros proved that if q = 1 and t ∈ [0, 1] is fixed, then S1(t) = T1(t) = 2T (t) is the smallest possible. In[25] Tabor and Tabor showed that if 1 ≤ q ≤ 2 and t ∈ [0, 1] is fixed, then Sq(t) is the smallest possible value sothat (9) be valid for all (ε, q)-Jensen convex functions f on D.For x, y ∈ D denote by [x, y] = {tx+ (1− t)y | t ∈ [0, 1]}. It is an important question whether the error termsTα(t, ‖x− y‖), Sα(t, ‖x− y‖) in (5) in (6) and Tq(t) in (8) are the smallest possible ones. In other words, for allfixed x, y ∈ D, we want to obtain the exact upper bound of the convexity-difference of α-Jensen convex functionsdefined by(10) Cα(x, y, t) := supf∈JCα(D){f(tx+ (1− t)y)− tf(x)− (1− t)f(y)},whereJCα(D) := {f : D → R |f is α-Jensen convex on D}.The statement of Theorem 1, Theorem 2, Corollary 3, and Corollary 4 can be stated as(11) Cα(x, y, t) ≤ τ(t, ‖x− y‖),where τ : R×D+ → R+ is given byτ := Tα, τ := Sα, τ(t, u) := εTq(t)uq, and τ(t, u) := εSq(t)uq,respectively. To obtain also a lower bound for Cα(x, y, t), (and thus to prove the sharpness of the inequality(11)), the following important observation was done by Páles in [23].Theorem 5. (Páles [23])Let α : D+ → R be continuous. Let τ : R×D+ → R be continuous in its first variable, with τ(0, u) = τ(1, u) = 0for all u ∈ D+, which is Jensen convex in the following sense, for all u ∈ D+ and s, t ∈ [0, 1],τ( t+ s2, u)≤ τ(t, u) + τ(s, u)2+ α(|t− s|u).Then,Cα(x, y, t) ≥ τ(t, ‖x− y‖)2. From strong α-Jensen convexity to strong convexitySimilarly as in Theorem 1 and Theorem 2, it can be proved two BernsteinDoetsch type results for locallyupper bounded strongly Jensen convex functions. Thus, these theorems give us connections between strongα-Jensen convexity and convexity type inequalities.Theorem 6. Let f : D → R be locally upper bounded on D and let α : D+ → R+. Then f is strongly α-Jensenconvex on D if and only if(12) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y)− Tα(t, ‖x− y‖)for all x, y ∈ D and t ∈ [0, 1].Theorem 7. Let f : D → R be upper semicontinuous on D and let α : D+ → R+ be∑∞n=n0α(2−n) < ∞ for somen0 ∈ N. Then f is α-Jensen convex on D if and only if(13) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y)− Sα(t, ‖x− y‖)for all x, y ∈ D and t ∈ [0, 1].We can also look for the optimal Takagi type function. In other words, for all fixed x, y ∈ D, we want toobtain the exact upper bound of the convexity-difference of strongly α-Jensen convex functions defined by(14) SCα(x, y, t) := supf∈SJCα(D){f(tx+ (1− t)y)− tf(x)− (1− t)f(y)},whereSJCα(D) := {f : D → R |f is locally upper bounded and strongly α-Jensen convex on D}.By Theorem 5, it is enough to prove the Jensen-convexity of Sα(·, u) or Tα(·, u). We shall prove that the Takagitype function Sα(·, u) will be the optimal choice. To show this suspicion let introduce the following Takagi typefunction Sϕ : [0, 1]→ R defined by(15) Sϕ(x) =∞∑n=0ϕ(12n)dZ(2nx),where P := {1, 12 , 14 , . . . , 12n , . . . , } and ϕ : P → R+ is a nonnegative function. The main results of this sectionstate that, under certain assumptions on the function ϕ : P → R, (−Sϕ) is well-defined and strongly Jensenconvex in the following sense: For all x, y ∈ [0, 1],(16) − Sϕ(x+ y2)≤ −Sϕ(x)− Sϕ(y)2− ϕ ◦ dZ(x− y2).First we describe the situation when the definition of Sϕ is correct.Lemma 8. Let ϕ : P → R+ be a nonnegative function. Then Sϕ is well-defined, i.e., the series on the right handside of (15) is convergent everywhere if and only if(17)∞∑n=0ϕ(12n)<∞.In the sequel, the class of nonnegative functions ϕ : P → R+ satisfying the condition (17) will be denoted byH:H :={ϕ : P → R+ |∞∑n=0ϕ(12n)<∞}.The next theorem, which was discovered by Jacek Tabor and Józef Tabor, has an important role in the proofof the main theorem of this section.Theorem 9. For every x, y ∈ RS2(x+ y2)≤ S2(x) + S2(y)2+ d2Z(x− y2).The following simple observation is a direct consequence of the previous theorem.Corollary 10. For every x, y ∈ [0, 1]− S2(x+ y2)=−S2(x)− S2(y)2− d2Z(x− y2).In the next result we give a representation of Sϕ(x) as an infinite linear combination of the values S2(2nx),n = 1, 2, . . . .Theorem 11. Let ϕ ∈ H. Then, for every x ∈ R,(18) Sϕ(x) = ϕ(1)S2(x) +∞∑n=1(ϕ(12n)− 14ϕ( 12n−1 ))S2(2nx).An immediate consequence of the previous two theorems is the next result which states the strong convexityof (−Sϕ).Theorem 12. Let ϕ ∈ H such that, for all u ∈ 12P , ϕ(2u) ≥ 4ϕ(u). Then, for all x, y ∈ R,(19) − Sϕ(x+ y2)≤ −Sϕ(x)− Sϕ(y)2− Φ2(x− y2),where Φ2 : R→ R is defined by(20) Φ2(u) :=∞∑n=0ϕ(12n)(d2Z(2nu)− 14d2Z(2n+1u)).In the next proposition we describe a decomposition property of the function Φ2.Proposition 13. For ϕ ∈ H, for all u ∈]0, 12 ],(21) Φ2(u) = Φ2( 12[log21u ]− u)+ ϕ( 12[log21u ]−1)(1− 2 · 2[log2 1u ]u).In the next proposition an important class of functions ϕ from H will be described.Proposition 14. Let ϕ : [0, 1] → R+. Assume that ϕ(0) = 0 and the mapping x 7→ ϕ(x)x is convex on ]0, 1]. Thenϕ|P ∈ H, the function x 7→ ϕ(x)x2 is nondecreasing on ]0, 1] and ϕ is continuous on [0, 1[.The next theorem has an important role in the proof of our subsequent main results.Theorem 15. Let ϕ : [0, 1]→ R+. Assume that ϕ(0) = 0 and the mapping x 7→ ϕ(x)x is convex on ]0, 1], then, for allu ∈ R,(22) − Φ2(u) ≤ −ϕ ◦ dZ(u).The main result of this section is stated in the following theorem.Theorem 16. Let ϕ : [0, 1]→ R+. Assume that ϕ(0) = 0 and the mapping x 7→ ϕ(x)x is convex on ]0, 1]. Then Sϕ isapproximately Jensen convex in the sense of (16).We shall prove that the error terms −Sα(t, ‖x− y‖) in (6) under certain assumptions on the error function αis the smallest possible one. In other words, the next theorem will provide exact upper bound for the convexity-difference of strongly α-Jensen convex functions defined by (14).Theorem 17. Let α : D+ → R be an error function such that α(0) = 0 and the map u 7→ ϕ(u)u is convex on D+ \{0}.Then, for all x, y ∈ D and t ∈ [0, 1],(23) SCα(x, y, t) = −Sα(t, ‖x− y‖).Taking an error function α which is a combination of power functions of exponents from [2,∞[, we obtain thefollowing result.Theorem 18. Let ν be a nonnegative bounded Borel measure on [2,∞[. Define the error function αν : D+ → R+ byαν(u) :=∫[2,∞[uqdν(q) (u ∈ D+).Then, for all x, y ∈ D and t ∈ [0, 1],SCαν (x, y, t) = −∫[2,∞[Sq(t)‖x− y‖qdν(q),where Sq : R→ R is given by (7).Corollary 19. Let q ∈ [2,∞[ and ε ≥ 0. Define the error function α : D+ → R+ by α(u) := εuq. Then, for allx, y ∈ D and t ∈ [0, 1],SCα(x, y, t) = −εSq(t)‖x− y‖q.The next figures demonstrate the strong convexity of −Sq, when q = 2 and q = 4.3. On a strong convexity type inequalityGiven a nonnegative even function α : D∗ → R+, we say that a map f : D → R is strongly α-convex, if for allx, y ∈ D and t ∈ [0, 1],(24) f(tx+ (1− t)y) ≤ tf(x) + (1− t)f(y)− tα((1− t)‖x− y‖)− (1− t)α(t‖y − x‖)holds. If (24) holds with T = {1/2}, i.e., for all x, y ∈ D,f(x+ y2)≤ f(x) + f(y)2− α(x−y2 ),we can get the strong α( ·2 )-Jensen convexity of the function f By the nonnegativity of α, we have that stronglyα-Jensen convex and strongly α-convex functions are always convex in the same sense, respectively.In [12], we examined the strong α-Jensen convexity and we got the following result.Theorem 20. (Makó-Nikodem-Páles, [12])For any function f : D → R, the following conditions are equivalent:(i) f is strongly α-convex.(ii) f is directionally differentiable at every point of D, and for all x0 ∈ D, the map h 7→ f ′(x0, h) is sublinearon X, furthermore for all x0, x ∈ D,(25) f(x) ≥ f(x0) + f ′(x0, x− x0) + α(‖x− x0‖).(iii) For all x0 ∈ D, there exits an element A ∈ X ′ such that(26) f(x) ≥ f(x0) +A(x− x0) + α(‖x− x0‖) for all x ∈ D.Thus, it can be important to look for connections between the strong α-Jensen convexity and strong α-convexity.Theorem 21. If f : D → R is locally upper bounded and strongly α-Jensen convex, then f is strongly 2α-convex onD.In the following theorems, we have established relations between HermiteHadamard type inequalities andstrong (Jensen) convexity.Theorem 22. Let ρ : [0, 1] → R be integrable function and assume that α : D∗ → R+ be a given error function.Denote λ :=∫ 10ρ. If f : D → R is continuous and satisfies the following upper HermiteHadamard type inequality∫ 10f(tx+ (1− t)y)ρ(t)dt ≤ λf(x) + (1− λ)f(y)− α(‖x− y‖), (x, y ∈ D)then f is strongly 1λα-convex on D.Theorem 23. Let ρ : [0, 1] → R be integrable function and assume that α : D∗ → R+ be a given error function.Denote λ :=∫ 10ρ. If f : D → R is continuous and satisfies the following lower HermiteHadamard type inequalityf(λx+ (1− λ)y) ≤∫ 10f(tx+ (1− t)y)ρ(t)dt− α(‖x− y‖) (x, y ∈ D)then f satisfies the following Jensen-type inequalityf(λx+ (1− λ)y) ≤ λf(x) + (1− λ)f(y)− α(‖x− y‖) (x, y ∈ D).References[1] A. Azócar and J. Giménez and K. Nikodem and J.L. Sánchez On sztongly midconvex functions Opuscula Mathematica 31:1526,2011.[2] F. Bernstein and G. Doetsch. Zur Theorie der konvexen Funktionen. Math. Ann., 76(4):514526, 1915.[3] Z. Boros. An inequality for the Takagi function. Math. Inequal. Appl., 11(4):757765, 2008.[4] A. Házy. On approximate t-convexity. Math. Inequal. 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On Strongly Convex Functions

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