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Research article Special Issues

Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations

  • Received: 27 September 2022 Revised: 27 February 2023 Accepted: 06 March 2023 Published: 17 March 2023
  • MSC : 06D72, 60Lxx, 60L70

  • The quantale module introduced by Abramsky and Vickers, engaged a large number of researchers. This research article focuses the combined behavior of rough set, soft set and an algebraic structure quantale module with the left action. In fact, the paper reflects the generalization of rough soft sets. This combined effect is totally dependent on soft binary relation including aftersets and foresets. Different soft substructures in quantale modules are defined. The characterizations of soft substructures in quantale modules based on soft binary relation are presented. Further, in quantale modules, we define soft compatible and soft complete relations in terms of aftersets and foresets. Furthermore, we use soft compatible and soft complete relations to approximate soft substructures of quantale modules and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough soft quantale modules. Additionally, we describe the algebraic relationships between the upper (lower) approximations of soft substructures of quantale modules and the upper (lower) approximations of their homomorphic images using the concept of soft quantale module homomorphism.

    Citation: Saqib Mazher Qurashi, Ferdous Tawfiq, Qin Xin, Rani Sumaira Kanwal, Khushboo Zahra Gilani. Different characterization of soft substructures in quantale modules dependent on soft relations and their approximations[J]. AIMS Mathematics, 2023, 8(5): 11684-11708. doi: 10.3934/math.2023592

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  • The quantale module introduced by Abramsky and Vickers, engaged a large number of researchers. This research article focuses the combined behavior of rough set, soft set and an algebraic structure quantale module with the left action. In fact, the paper reflects the generalization of rough soft sets. This combined effect is totally dependent on soft binary relation including aftersets and foresets. Different soft substructures in quantale modules are defined. The characterizations of soft substructures in quantale modules based on soft binary relation are presented. Further, in quantale modules, we define soft compatible and soft complete relations in terms of aftersets and foresets. Furthermore, we use soft compatible and soft complete relations to approximate soft substructures of quantale modules and these approximations are interpreted by aftersets and foresets. This concept generalizes the concept of rough soft quantale modules. Additionally, we describe the algebraic relationships between the upper (lower) approximations of soft substructures of quantale modules and the upper (lower) approximations of their homomorphic images using the concept of soft quantale module homomorphism.



    Ostrowski's Inequality. Let f:I[0,+)R be a differentiable function on int(I), such that fL[a,b], where a,bI with a<b. If |f(x)|M for all x[a,b], then the inequality:

    |f(x)1babaf(t)dt|M(ba)[14+(xa+b2)2(ba)2],     x[a,b] (1.1)

    holds for all x[a,b]. This inequality was introduced by Alexander Ostrowski in [26], and with the passing of the years, generalizations on the same, involving derivatives of the function under study, have taken place. It is playing a very important role in all the fields of mathematics, especially in the theory approximations. Thus such inequalities were studied extensively by many researches and numerous generalizations, extensions and variants of them for various kind of functions like bounded variation, synchronous, Lipschitzian, monotonic, absolutely continuous and n-times differentiable mappings etc.

    For recent results and generalizations concerning Ostrowski's inequality, we refer the reader to the recent papers [1,3,4,31,32]. The convex functions play a significant role in many fields, for example in biological system, economy, optimization and so on [2,16,19,24,29,34,39]. And many important inequalities are established for these class of functions. Also the evolution of the concept of convexity has had a great impact in the community of investigators. In recent years, for example, generalized concepts such as s-convexity (see[10]), h-convexity (see [30,33]), m-convexity (see [7,15]), MT-convexity (see[21]) and others, as well as combinations of these new concepts have been introduced.

    The role of convex sets, convex functions and their generalizations are important in applied mathematics specially in nonlinear programming and optimization theory. For example in economics, convexity plays a fundamental role in equilibrium and duality theory. The convexity of sets and functions have been the object of many studies in recent years. But in many new problems encountered in applied mathematics the notion of convexity is not enough to reach favorite results and hence it is necessary to extend the notion of convexity to the new generalized notions. Recently, several extensions have been considered for the classical convex functions such that some of these new concepts are based on extension of the domain of a convex function (a convex set) to a generalized form and some of them are new definitions that there is no generalization on the domain but on the form of the definition. Some new generalized concepts in this point of view are pseudo-convex functions [22], quasi-convex functions [5], invex functions [17], preinvex functions [25], B-vex functions [20], B-preinvex functions [8], E-convex functions [38], Ostrowski Type inequalities for functions whose derivatives are (m,h1,h2)-convex [35], Féjer Type inequalities for (s,m)-convex functions in the second sense [36] and Hermite-Hadamard-Féjer Type inequalities for strongly (s,m)-convex functions with modulus C, in the second sense [9]. In numerical analysis many quadrature rules have been established to approximate the definite integrals. Ostrowski inequality provides the bounds of many numerical quadrature rules [13].

    In this paper we have established new Ostrowski's inequality given by Badreddine Meftah in [23] for s-φ-convex functions with fCn([a,b]) such that f(n)L([a,b]) and we give some applications to some special means, the midpoint formula and some examples for the case n=2.

    Recall that a real-valued function f defined in a real interval J is said to be convex if for all x,yJ and for any t[0,1] the inequality

    f(tx+(1t)y)tf(x)+(1t)f(y) (2.1)

    holds. If inequality 2.1 is strict when we say that f is strictly convex, and if inequality 2.1 is reversed the function f is said to be concave. In [37] we introduced the notion of s-φ-convex functions as a generalization of s-convex functions in first sense.

    Definition 1. Let 0<s1. A function f:IRR is called s-φ-convex with respect to bifunction φ:R×RR (briefly φ-convex), if

    f(tx+(1t)y)f(y)+tsφ(f(x),f(y)) (2.2)

    for all x,yI and t[0,1].

    Example 1. Let f(x)=x2, then f is convex and 12-φ- convex with φ(u,v)=2u+v, indeed

    f(tx+(1t)y)=(tx+(1t)y)2=t2x2+2t(1t)xy+(1t)2y2y2+tx2+2txy=y2+t12[t12x2+2t12xy].

    On the other hand;

    0<t<10<t12<1t12x2+2t12xyx2+2xyx2+x2+y2.

    Hence,

    f(tx+(1t)y)y2+t12[2x2+y2]=f(y)+t12φ(f(x),f(y)).

    Example 2. Let f(x)=xn and 0<s1, then f is convex and s-φ- convex with φ(u,v)=nk=1(nk)v1kn(u1nv1n)n, indeed

    f(tx+(1t)y)=f(y+t(xy))=(y+t(xy))n=yn+nk=1(nk)ynk(t(xy))n=yn+ts[nk=1(nk)tnsynk(xy)n]yn+ts[nk=1(nk)(yn)nkn((xn)1n(yn)1n)n].

    Remark 1. If f is increasing monotone in [a,b], then f is s-φ- convex for φ(x,y)=K, where K[0,+) and s(0,1].

    In this section, we give some integral approximation of fCn([a,b]) such that f(n)L([a,b]), for n1 using the following lemma as the main tool (see [11]).

    Lemma 1. Let f:[a,b]R be a differentiable mapping such that f(n1) is absolutely continuous on [a,b]. Then for all x[a,b] we have the identity

    baf(t)dt=nk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)+(1)nbaKn(x,t)f(n)(t)dt,

    where the kernel Kn:[a,b]2R is given by

    Kn(x,t)={(ta)nn!ift[a,x](tb)nn!ift(x,b]

    with x[a,b] and n is natural number, n1.

    Theorem 1. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1 and 0<s1. If |f(n)| is s-φ-convex, then the following inequality

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1n!(1n+1|f(n)(a)|+1n+s+1φ(|f(n)(a)|,|f(n)(x)|))+(bx)n+1n![|f(n)(x)|n+1+nk=0(nk)(1)k1k+s+1φ(|f(n)(x)|,|f(n)(b)|)]

    holds for all x[a,b].

    Proof. From Lemma 1, properties of modulus, making the changes of variables u=(1t)a+tx in the first integral and u=(1t)x+tb in the second integral we have that,

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|xa(ua)nn!|f(n)(u)|du+bx(bu)nn!|f(n)(u)|du=(xa)n+1n!10tn |f(n)((1t)a+tx)|dt+(bx)n+1n!10(1t)n |f(n)((1t)x+tb)|dt.

    Since |f(n)| is s-φ- convex (2.2) gives

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1n!10tn(|f(n)(a)|+tsφ(|f(n)(a)|,|f(n)(x)|))dt+(bx)n+1n!10(1t)n(|f(n)(x)|+tsφ(|f(n)(x)|,|f(n)(b)|))dt=(xa)n+1n!(1n+1|f(n)(a)|+1n+s+1φ(|f(n)(a)|,|f(n)(x)|))+(bx)n+1n![|f(n)(x)|n+1+nk=0(nk)(1)k1k+s+1φ(|f(n)(x)|,|f(n)(b)|)]

    which is the desired result. The proof is completed.

    Remark 2. If we take s=1 then obtain a result of Meftah B. (see Theorem 2.1 in [23]).

    Corollary 1. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1 and 0<s1. If |f(n)| is s-convex in the first sense, we have the following estimate

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|s(n+1)!(n+s+1)|f(n)(a)|+(bx)n+1n!|f(n)(b)|nk=0(nk)(1)kk+s+1+(n+1)[(xa)n+1(n+s+1)(n+1)!+(bx)n+1(n+1)!(1n+1nk=0(nk)(1)kk+s+1)]|f(n)(x)|.

    Proof. Taking φ(u,v)=vu in Theorem 1.

    Remark 3. It is important to notice that if s=1 we have that |f(n)| is convex and then obtain the corollary 2.2 of Meftah see [23].

    Theorem 2. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1 and let q>1 with 1p+1q=1. If |f(n)|q is s-φ-convex, then the following inequality holds

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(a)|q+φ(|f(n)(a)|q,|f(n)(x)|q))1q+(bx)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q))1q.

    Proof. From Lemma 1, properties of modulus, and Holder's inequality, we have

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|xa(ua)nn!|f(n)(u)|du+bx(bu)nn!|f(n)(u)|du=(xa)n+1n!10tn |f(n)((1t)a+tx)|dt+(bx)n+1n!10(1t)n |f(n)((1t)x+tb)|dt(xa)n+1n!(10tnpdt)1p(10|f(n)((1t)a+tx)|qdt)1q+(bx)n+1n!(10(1t)npdt)1p(10|f(n)((1t)x+tb)|qdt)1q=(xa)n+1(np+1)1pn!(10|f(n)((1t)a+tx)|qdt)1q+(bx)n+1(np+1)1pn!(10|f(n)((1t)x+tb)|qdt)1q.

    Since |f(n)|q is s-φ-convex, we deduce

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1(np+1)1pn!(10(|f(n)(a)|q+tsφ(|f(n)(a)|q,|f(n)(x)|q))dt)1q+(bx)n+1(np+1)1pn!(10(|f(n)(x)|q+tsφ(|f(n)(x)|q,|f(n)(b)|q))dt)1q
    =(xa)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(a)|q+φ(|f(n)(a)|q,|f(n)(x)|q))1q+(bx)n+1(s+1)1q(np+1)1pn!((s+1)|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q))1q.

    Corollary 2. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1 and let q>1 with 1p+1q=1. If |f(n)|q is s-convex in the first sense, then the following inequality holds

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1(s+1)1q(np+1)1pn!(s|f(n)(a)|q+|f(n)(x)|q)1q+(bx)n+1(s+1)1q(np+1)1pn!(s|f(n)(x)|q+|f(n)(b)|q)1q. (3.1)

    Proof. Taking φ(u,v)=vu in Theorem 1.

    Corollary 3. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1 and let q>1 with 1p+1q=1. If |f(n)|q is s-convex in the first sense, then the following inequality holds

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1(s+1)1q(np+1)1pn!(s|f(n)(a)|+|f(n)(x)|)+(bx)n+1(s+1)1q(np+1)1pn!(s|f(n)(x)|+|f(n)(b)|).

    Proof. Taking φ(u,v)=vu in Theorem 1, we obtain 3.1. Then using the following algebraic inequality for all a,b0, and 0α1 we have (a+b)αaα+bα, we get the desired result.

    Theorem 3. Let q>1 and f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1. If |f(n)|q is s-φconvex, then the following inequality

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(n+1)1q(xa)n+1(n+1)!(1n+1|f(n)(a)|q+1n+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(n+1)1q(bx)n+1(n+1)!(1n+1|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q)nk=0(nk)(1)kk+s+1)1q

    holds for all x[a,b].

    Proof. From Lemma 1, properties of modulus, and power mean inequality, we have

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|xa(ua)nn!|f(n)(u)|du+bx(bu)nn!|f(n)(u)|du=(xa)n+1n!10tn |f(n)((1t)a+tx)|dt+(bx)n+1n!10(1t)n |f(n)((1t)x+tb)|dt
    (xa)n+1n!(10tndt)11q(10tn |f(n)((1t)a+tx)|qdt)1q+(bx)n+1n!(10(1t)ndt)11q(10(1t)n |f(n)((1t)x+tb)|qdt)1q=(n+1)1q(xa)n+1(n+1)!(10tn |f(n)((1t)a+tx)|qdt)1q+(n+1)1q(bx)n+1(n+1)!(10(1t)n |f(n)((1t)x+tb)|qdt)1q.

    Since |f(n)|q is s-φ-convex, we deduce

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(n+1)1q(xa)n+1(n+1)!(|f(n)(a)|q10tndt+φ(|f(n)(a)|q,|f(n)(x)|q)10tn+sdt)1q+(n+1)1q(bx)n+1(n+1)!(|f(n)(x)|q10(1t)ndt+φ(|f(n)(x)|q,|f(n)(b)|q)10ts(1t)ndt)1q=(n+1)1q(xa)n+1(n+1)!(1n+1|f(n)(a)|q+1n+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(n+1)1q(bx)n+1(n+1)!(1n+1|f(n)(x)|q+φ(|f(n)(x)|q,|f(n)(b)|q)nk=0(nk)(1)kk+s+1)1q.

    The proof is completed.

    Remark 4. If we take s=1 then obtain a result of Meftah B. (see Theorem 2.6 in [23]).

    Corollary 4. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1 and let q>1. If |f(n)|q is s-convex in the first sense, then the following inequality

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(n+1)1q(xa)n+1(n+s+1)1q(n+1)!(s|f(n)(a)|qn+1+|f(n)(x)|q)1q+(n+1)1q(bx)n+1(n+1)!(1n+1|f(n)(x)|q+[|f(n)(b)|q|f(n)(x)|q]nk=0(nk)(1)kk+s+1)1q

    holds for all x[a,b].

    Proof. Taking φ(u,v)=vu in Theorem 3.

    Theorem 4. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1 and let q>1. If |f(n)|q is s-φ-convex, then the following inequality

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1n!(1qn+1|f(n)(a)|q+1qn+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(bx)n+1n!(1qn+1|f(n)(x)|q+qnk=0(qnk)(1)kk+s+1φ(|f(n)(x)|q,|f(n)(b)|q))1q

    holds for all x[a,b].

    Proof. From Lemma 1, properties of modulus, and power mean inequality, we have

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|xa(ua)nn!|f(n)(u)|du+bx(bu)nn!|f(n)(u)|du=(xa)n+1n!10tn |f(n)((1t)a+tx)|dt+(bx)n+1n!10(1t)n |f(n)((1t)x+tb)|dt(xa)n+1n!(10dt)11q(10tqn |f(n)((1t)a+tx)|qdt)1q+(bx)n+1n!(10dt)11q(10(1t)qn |f(n)((1t)x+tb)|qdt)1q=(xa)n+1n!(10tqn |f(n)((1t)a+tx)|qdt)1q+(bx)n+1n!(10(1t)qn |f(n)((1t)x+tb)|qdt)1q.

    Since |f(n)|q is s-φ-convex, we deduce

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1n!(|f(n)(a)|q10tqndt+φ(|f(n)(a)|q,|f(n)(x)|q)10tqn+sdt)1q+(bx)n+1n!(|f(n)(x)|q10(1t)qndt+φ(|f(n)(x)|q,|f(n)(b)|q)10ts(1t)qndt)1q
    =(xa)n+1n!(1qn+1|f(n)(a)|q+1qn+s+1φ(|f(n)(a)|q,|f(n)(x)|q))1q+(bx)n+1n!(1qn+1|f(n)(x)|q+qnk=0(qnk)(1)kk+s+1φ(|f(n)(x)|q,|f(n)(b)|q))1q

    which in the desired result.

    Remark 5. If we take s=1 then obtain a result of Meftah B. (see Theorem 2.9 in [23]).

    Corollary 5. Let f:IR be n-times differentiable on [a,b] such that f(n)L([a,b]) with n1, 0<s1 and let q>1. If |f(n)|q is s-convex in the first sense, then the following inequality

    |baf(t)dtnk=0[(bx)k+1+(1)k(xa)k+1(k+1)!]f(k)(x)|(xa)n+1n!(1qn+1|f(n)(a)|q+|f(n)(x)|q|f(n)(a)|qqn+s+1)1q+(bx)n+1n!(1qn+1|f(n)(x)|q+qnk=0(qnk)(1)kk+s+1(|f(n)(b)|q|f(n)(x)|q))1q

    holds for all x[a,b].

    Proof. Taking φ(u,v)=vu in Theorem 4.

    In this section, using [12] we define s-φb-convex function as generalized form of s-φ convex functions [37] and give some results.

    Definition 2. Let R+ be the set of nonnegative real numbers and b:R×R×[0,1]R+ be a function with tsb(x,y,t)[0,1] for all x,yR, t[0,1] and s(0,1]. A function f:IR is called s-φb-convex if

    f(tx+(1t)y)f(y)+tsb(x,y,t)φ(f(x),f(y))

    for all x,yR and t[0,1].

    Remark 6. If b(x,y,z)=1 then the definition of s-φb-convex function matches the definition of s-φ-convex function.

    Theorem 5. Consider a function f:IR and b:R×R×[0,1]R+ be a function with tsb(x,y,t)[0,1] for all x,yR and s,t[0,1]. Then the following assertions are equivalent:

    (i) f is s-φb-convex for some b and s[0,1].

    (ii) f is φ-quasiconvex.

    Proof. (i)(ii) For any x,yI and t[0,1],

    f(tx+(1t)y)f(y)+tsb(x,y,t)φ(f(x),f(y))max{f(y),f(y)+φ(f(x),f(y))}.

    (ii)(i) For x,yI and t[0,1], define

    b(x,y,t)={1ts if t[0,1] and f(y)f(y)+φ(f(x),f(y))0 if t=0 or f(y)>f(y)+φ(f(x),f(y))

    Notice that tsb(x,y,t)[0,1]. For a such function b we have

    f(tx+(1t)y)max{f(y),f(y)+φ(f(x),f(y))}=tsb(x,y,t)[f(y)+φ(f(x),f(y))+(1tsb(x,y,t))]f(y)=f(y)+tsb(x,y,t)φ(f(x),f(y)).

    Remark 7. Let f:IR be a s-φ-convex function. For x1,x2I and α1+α2=1, we have f(α1x1+α2x2)f(x2)+αs1φ(f(x1),f(x2)). Aso when n>2, for x1,x2,...,xnI, ni=1αi=1 and Ti=ij=1αj, we have

    f(ni=1αixi)=f((Tn1n1i=1αiTn1xi)+αnxn)f(xn)+Tsn1φ(f(n1i=1αiTn1xi),f(xn)). (4.1)

    Theorem 6. Let f:IR be a s-φ-convex function and φ be nondecreasing nonnegatively sublinear in first variable. If Ti=ij=1αj for i=1,2,...,n such that Tn=1, then

    f(ni=1αixi)f(xn)+n1i=1Tsiφf(xi,xi+1,...,xn),

    where φf(xi,xi+1,...,xn)=φ(φf(xi,xi+1,...,xn1),f(xn)) and φf(x)=f(x) for all xI.

    Proof. Since φ is nondecreasing nonnegatively sublinear on first variable, so from (4.1) it follows that:

    f(ni=1αixi)=f((Tn1n1i=1αiTn1xi)+αnxn)f(xn)+Tsn1φ(f(n1i=1αiTn1xi),f(xn))=f(xn)+(Tn1)sφ(f(Tn2Tn1n2i=1αiTn2xi+αn1Tn1xn1),f(xn))f(xn)+(Tn1)sφ(f(xn1)+(Tn2Tn1)sφ(f(n2i=1αiTn2xi),f(xn1)),f(xn))f(xn)+(Tn1)sφ(f(xn1),f(xn))+(Tn2)sφ(φ(f(n2i=1αiTn2xi),f(xn1)),f(xn))...
    f(xn)+(Tn1)sφ(f(xn1),f(xn))+(Tn2)sφ(φ(f(xn2),f(xn1)),f(xn))+...+Ts1φ(φ(...φ(φ(f(x1),f(x2)),f(x3)...),f(xn1)),f(xn))=f(xn)+(Tn1)sφf(xn1,xn)+(Tn2)sφf(xn2,xn1,xn)+...+(T1)sφf(x1,x2,...,xn1,xn)=f(xn)+n1i=1Tsiφf(xi,xi+1,...,xn).

    Example 3. Consider f(x)=x2 and φ(x,y)=2x+y for x,yR+=[0,+). The function φ is nondecreasing nonnegatively sublinear in first variable and f is 12-φ-convex (see Example 1). Now for x1,x2,...,xnR+ and α1,α2,...,αn with ni=1αi=1 according to Theorem 6 we have

    (ni=1αixi)2(xn)2+n1i=1T12iφf(xi,xi+1,...,xn)(xn)2+n1i=1T12i[2[...2[2x2i+x2i+1]+x2i+2]+...+x2n].

    In this section we give some applications for the special case where n=2 and the function φ(f(x),f(y))=f(y)f(x), in this case we have that f is s-convex in the first sense.

    Example 4. Let s(0,1) and p,q,rR, we define the function f:[0,+)R as

    f(t)={pift=0qts+rift>0,

    we have that if q0 and rp, then f is s-convex in the first sense (see [18]). If we do φ(f(x),f(y))=f(x)f(y), then f is s-φ-convex, but is not φ-convex because f is not convex.

    Example 5. In the previous example if s=12, p=1, q=2 and r=1 we have that f:[0,+)R,f(t)=2t12+1 is 12-φ-convex. Then if we define g:[0,+)R, g(t)=815t52+t22, we have to g(t)=2t12+1 is 12-φ-convex in [0,+) with φ(f(x),f(y))=f(x)f(y). Using Theorem 1, for a,b[0,+) with a<b and x[a,b], we get

    |16(b72a72)+35(a3b3)x+35(b2a2)x32+21(ab)x52|352(1+2b)(bx)3103(72+a+6b)(ax)3.

    Remark 6. In particular if we choose a=0 and b=1, we have for x[0,1], we get a graphic representation of the Example 5.

    Example 6. If we define g(t)=t412 we have that g(t) is 12-φ- convex with φ(u,v)=2u+v (see example 1) and by Theorem 1, for a,bR with a<b and x[a,b], we have that

    |b5a560(ba)12x4[b22x(ba)a23]x3[(bx)3+(xa)36]x2|(xa)3[x256a2]+(bx)3[19210x2+8105b2].

    Moreover, if choose x=a+b2, we obtain that

    |b5a560(ba)(a+b)4192(ba)3(a+b)296|(ba)316[a23+9a2+2ab+b214+(a+b)212+8a2+16ab+24b2105].

    Then

    |(ab)5|(ba)37(477a2+194ab+161b2).

    Therefore

    (ab)2477a2+194ab+161b27.

    Example 7. If we define g(t)=369132 t136 we have that |g(t)|3 is 12-φ-convex with φ(u,v)=2u+v (see example 1) and by Theorem 4, for a,bR with a<b and x[a,b], we have

    |2161729[b196a196]36x13691(ba)+6x7614[(bx)2(xa)2]+x166[(bx)3+(xa)3]|(xa)32348(a12+2x12)13+(bx)32348(x12+2b12)13.

    In this paper we have established new Ostrowski's inequality given by Badreddine Meftah in [23] for sφconvex functions with fCn([a,b]) such that f(n)L([a,b]) with n1 and we give some applications to some special means, the midpoint formula and some examples for the case n=2. We expect that the ideas and techniques used in this paper may inspire interested readers to explore some new applications of these newly introduced explore some new applications of these newly introduced functions in various fields of pure and applied sciences.

    The authors want to give thanks to the Dirección de investigación from Pontificia Universidad Católica del Ecuador for technical support to our research project entitled: "Algunas desigualdades integrales para funciones convexas generalizadas y aplicaciones".

    The authors declare that they have no conflicts of interest.



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