In this study, we give a new \(m\)-convex function that is called an \(m\)-convex of the second type and its some properties. Moreover, some integral inequalities are examined for each \(m\)-convex function of the second type.
The concept of a convex function is equivalent to the set of all points forming the convex function’s graph being convex [1]. Thus, problems involving convex functions are associated with convex sets. As a result of a thorough understanding of convex geometry, many mathematicians are better able to address problems from a geometric perspective.
Convex functions can occasionally be found in proof methods given in various mathematics topics. For example, Kittaneh used a simple property of convex functions to provide a better lower bound on the numerical radius of bounded linear operators. Consequently, the theory of convex functions remains a popular topic. Various classes of convex functions are studied by many mathematicians. One of these is the class of m-convex functions given by Toader, whose definition is given below [2].
A function \(f : [0,b] \to \mathbb{R}\) is called an \(m\)-convex function, where \(m \in (0,1]\) and \(b>0\), if for all \(x,y \in [0,b]\) and \(t \in [0,1]\) we have \[f\left(tx + m(1-t)y\right) \leq tf(x) + m(1-t)f(y).\]
If \(m=1\), then it is a convex function. If \(f(tx) \leq tf(x)\), then it is a star-shaped function.
The following relations are true for a non negative continuous function which vanishes at the origin [3] Convex functions \(\subset\) Star – shaped functions and also \(m\)-Convex functions \(\subset\) Star – shaped. Integral inequalities of \(m\)-convex functions have also been studied by some mathematicians [4–7]. Furthermore, inequalities satisfied by functions defined on linear operators can also be found in the literature [8,9].
On the other hand, Takahashi introduced the concept of convexity at a metric distance where the linear vector space conditions are not satisfied [10]. Fixed point theorems on these convex abstract metric spaces have become a subject of considerable study [11–14]. Moreover, the definition of \(m\)-convexity in \(b\)-metric spaces is defined by taking inspiration from \(m\)-convex functions in [15]. However, this definition corresponds to a very special case of convex metric spaces [16].
In this article, we adapt the definition given by Sertbaş et al. to functions [16]. First, we give some properties of these functions. Then, we calculate some basic integral inequalities.
Definition 1. \(f : [0,b] \to \mathbb{R}\), \(b>0\) is called to be an \(m\)– convex of the second type for some \(m \in (0,1]\) if \[f\left(tx + (1-t)my\right) \leq tf(x) + (1-mt)f(y),\] for any \(x,y \in [0,b]\) and \(t \in [0,1]\).
It is obvious that a non negative \(m\)-convex function and non negative continuous convex function which vanishes at the origin are an \(m\)-convex of the second type.
Theorem 1. If a function \(f : [0,b] \to \mathbb{R}\), \(b>0\) is a second type \(m\)-convex with \(m \in (0,1)\), then the following statements are satisfied;
i) \(f\) is a non negative function,
ii) The sequence \(\{f(m^n x)\}\), \(n \in \mathbb{N}\) converges for all \(x \in [0,b]\) and \[\lim_{n\to+\infty} f(m^n x) \leq \min\{f(0),f(x)\},\] is satisfied.
iii) \(f\) is continuous at zero if and only if \(f(0) \leq f(x)\) for each \(x \in [0,b]\).
Proof. In this case, for any \(x,y \in [0,b]\) \[f(y)=f(y+0mx) \leq f(y)+(1-m)f(x),\] and then \(f(x)\geq 0\) for each \(x\in[0,b]\). On the other hand, for any \(x\in[0,b]\) \[f(mx)=f(0+(1-0)mx) \leq f(x),\] is true. Therefore, \[0 \leq f(m^n x) \leq f(m^{n-1}x) \leq \cdots \leq f(mx) \leq f(x),\] is holds for \(n \in \mathbb{N}\). It means that the sequence \(\{f(m^n x)\}\) is monotone deceasing and bounded and so it converges. Also, for \(n \in \mathbb{N}\) \[f(m^n x)=f(m^n x+(1-m^n)0) \leq m^n f(x)+(1-m^{n+1})f(0),\] is correct. From these results \[\lim_{n\to+\infty} f(m^n x) \leq \min\{f(0),f(x)\},\] is obtained. Moreover, if \(f\) is continuous at zero, then from the last limit the inequality \(f(0)\leq f(x)\) is get for each \(x\in[0,b]\).
On the contrary, suppose that the inequality \(f(0)\leq f(x)\) is true for each \(x\in[0,b]\). In this case, for any \(x\in[0,b]\) there exist an element \(t\in[0,1]\) such that \(x=tb\) and \[ 0\leq f(x)-f(0)=f(tb)-f(0)\leq tf(b)+(1-mt)f(0)-f(0)=t[f(b)-mf(0)],\tag{1}\] is correct and so \(f\) continuous at zero. ◻
Corollary 1. If a function \(f : [0,b] \to \mathbb{R}\), \(b>0\) is an \(m\)-convex of the second type with \(m\in(0,1)\) and continuous at zero, then \[0 \leq \underline{\lim}_{x\to0}\frac{f(x)-f(0)}{x} \leq \overline{\lim}_{x\to0}\frac{f(x)-f(0)}{x} \leq \frac{f(b)-mf(0)}{b}.\]
Also, if \(f(x)\) is vanish at origin and continuous on \([0,b]\), then it is differentiable at \(x=0\).
Proof. According to previous theorem and the inequality (1) the first assertion can be proved.
If \(f(0)=0\), then it is a star shaped function. Therefore, it is obtained the second assertion by using Theorem 6 in [3]. ◻
Theorem 2. Assume that a function \(f : [0,b] \to \mathbb{R}\), \(b>0\) is an \(m\)-convex of the second type with \(m\in(0,1)\), then it is bounded on \([0,b]\) and following statements are hold
i) \(f\) is a bounded function on \([0,b]\),
ii) For any \(x\in[0,b]\), \(\overline{\lim}_{y\to x} f(my) \leq f(x)\).
Proof. For any \(x\in[0,b]\) there exist a unique element \(t\in[0,1]\) such that \(x=tb\) and \[0 \leq f(x)=f(tb)\leq tf(b)+(1-mt)f(0) \leq f(b)+f(0),\] this means that \(f\) is a bounded function on \([0,b]\).
On the other hand, \(\delta\) is a positive or negative number and small enough such that \(x+n\delta\in[0,b]\), \(n\in\mathbb{N}\). In this case, \[f(m(x+\delta))=f\left(\frac{m(x+n\delta)}{n}+\left(1-\frac{1}{n}\right)mx\right)\leq \frac{1}{n}f(m(x+n\delta))+ \left(1-\frac{m}{n}\right)f(x).\]
If \(\delta\to0\), then it must be \(n\to+\infty\). Since \(f\) is a bounded function, \[\overline{\lim}_{y\to x}f(my)\leq f(x),\] is get. This idea is in [17]. ◻
Theorem 3. Let \(f : [0,b] \to \mathbb{R}\), \(b>0\) be an \(m\)-convex of the second type and Lebesgue integrable on \([ma,c]\), \(0\leq a<c\leq b\). Then \[f\left(m\frac{a+c}{2}\right)\leq \frac{3-m}{2(c-a)}\int_a^c f(x)\,dx \leq (3-m)\left(\frac{c-ma}{c-a}\right)\frac{f(c)+(2-m)f(a)}{4}.\]
Proof. Because \(f\) is an \(m\)-convex of the second type a, for all \(x,y\in[a,b]\) we have \[f\left(m\frac{x+y}{2}\right) \leq \frac{f(x)+(2-m)f(y)}{2}.\]
If \(x=ta+(1-t)c\) and \(y=tc+(1-t)a\) are chosen, then we have \[f\left(m\frac{a+c}{2}\right)\leq \frac{(f(ta+(1-t)c)+(2-m)f(tc+(1-t)a))}{2}.\]
We obtain by integrating the last inequality \[f\left(m\frac{a+c}{2}\right)\leq \frac{3-m}{2(c-a)}\int_a^c f(x)\,dx.\]
On the other hand, \[\int_a^c f(y)\,dy \leq \int_{ma}^c f(y)\,dy \leq (c-ma)\frac{f(c)+(2-m)f(a)}{2}.\]
This is completed the proof of theorem. ◻
Theorem 4. Suppose that \(f : [0,b] \to \mathbb{R}\), \(b>0\) is a function and \(f’\) is an \(m\)-convex of the second type on \([0,mb]\), then for \(0\leq a<b\) \[\frac{f(mb)+(2-m)f(ma)}{3-m}-\frac{1}{(b-a)}\int_a^b f(mx)\,dx \leq \frac{m(b-a)}{3-m}\frac{(2-m) f'(b)+f'(a)}{4}.\]
Proof. Because \(f’\) is bounded on \([0,mb]\) and the points of discontinuity of a derivative function are only of the second type, \(f’\) must be continuous on \([0,mb]\). In this case, we use the idea given in [18], and so \[\int_0^1 (1-(3-m)t)f'(tma+(1-t)mb)\,dt = \frac{(2-m)f(ma)+f(mb)}{m(b-a)}- \frac{3-m}{m(b-a)^2}\int_a^b f(mx)\,dx,\] is get. Because \(f’\) is an \(m\)-convex of the second type on \([0,mb]\), \[\begin{aligned} \int_0^1 &(1-(3-m)t)f'(tma+(1-t)mb)\,dt \leq \int_0^1 (1-2t)f'(tma+(1-t)mb) \\ &\leq \int_0^1 |1-2t|f'(tma+(1-t)mb)\,dt \leq \int_0^1 |1-2t|\bigl(tf'(a)+(1-mt)f'(b)\bigr)\,dt \\ &=\frac{(2-m) f'(b)+f'(a)}{4}. \end{aligned}\]
From this inequality, \[\frac{f(mb)+(2-m)f(ma)}{3-m}-\frac{1}{(b-a)}\int_a^b f(mx)\,dx \leq \frac{m(b-a)}{3-m}\frac{(2-m) f'(b)+f'(a)}{4}.\] ◻
Corollary 2. Under the previous theorem, the following inequality is obtained \[f(ma)-\frac{1}{(b-a)}\int_a^b f(mx)\,dx < (b-a)\frac{(2-m) f'(b)+f'(a)}{8},\]
Proof. Since \(f’\) is a non negative continuous function on \([0,mb]\), \(f(x)=f(0)+\int_0^x f'(\tau)d\tau\) is a monotone increasing function. Also, \(\frac{m}{3-m}<\frac{1}{2}\) is true for \(m<1\). In this case, the desired result from the previous theorem is obtained. ◻
Roberts, A. W., & Varberg, D. E. (1974). Convex Functions: Convex Functions (Vol. 57). Academic Press.
Toader, G. (1984, October). Some generalizations of the convexity. In Proceedings of the Colloquium on Approximation and Optimization (Vol. 329, p. 338). Cluj-Napoca: University of Cluj-Napoca.
Bruckner, A. M., & Ostrow, E. (1962). Some function classes related to the class of convex functions. Pacific Journal of Mathematics, 12(4), 1203-1215.
Dragomir, S. S., & Toader, G. H. (1993). Some inequalities for m-convex functions, Studia Univ. Babes-Bolyai Math, 38(1), 21-28.
Bakula, M. K., Ozdemir, M. E., & Pecaric, J. (2008). Hadamard type inequalities for m-convex and \((\alpha, m)\)-convex functions. Journal of Inequalities in Pure and Applied Mathematics, 9(4), 12.
Pavic, Z., & Ardic, M. A. (2017). The most important inequalities of \(m\)-convex functions. Turkish Journal of Mathematics, 41(3), 625-635.
Sertbaş, M., & Mihyaz, İ. (2022). Some Results for \((s, m)\)-convex Function in the Second Sense. Maltepe Journal of Mathematics, 4(1), 1-8.
Erdaş, Y., Ünlüyol, E., & Salaş, S. (2019). Some New Inequalities of Operator m-Convex Functions and Applications for Synchronous–Asynchronous Functions. Complex Analysis and Operator Theory, 13(8), 3871-3881.
Maulana, E., Karim, C., & Kurniawaty, M. (2025). Exploring the \((h, m)\)-Convexity for Operators in Hilbert Space. CAUCHY: Jurnal Matematika Murni Dan Aplikasi, 10(1), 362-373.
Takahashi, W. (1970). A convexity in metric space and nonexpansive mappings, I. In Kodai Mathematical Seminar Reports (Vol. 22, No. 2, pp. 142-149). Department of Mathematics, Tokyo Institute of Technology.
Hammack, R. H., Imrich, W., Klavžar, S., Imrich, W., & Klavžar, S. (2011). Handbook of Product Graphs (Vol. 2). Boca Raton: CRC press.
Beg, I. (2025). Separating hyperplane theorems in convex metric spaces. Journal of Nonlinear Sciences and Applications, 18, 29-32.
Chen, L., Li, C., Kaczmarek, R., & Zhao, Y. (2020). Several fixed point theorems in convex \(b\)-metric spaces and applications. Mathematics, 8(2), 242.
Zhang, G., Shi, Y., & Yao, W. (2025). Convexity structures of the Hausdorff fuzzy quasi-metric spaces. Information Sciences, 718, 122406.
Amirbostaghi, G., Asadi, M., & Mardanbeigi, M. R. (2021). \(m\)-Convex Structure on \(b\)-Metric Spaces. Filomat, 35(14), 4765-4776.
Sertbaş, M., Büyükkaya, A., Öztürk, M., Existence and Stability Results for Fractional Boundary Value Problems in m-Convex b-Metric Spaces (preprint).
Hardy, G. H., Littlewood, J. E., & Pólya, G. (1952). Inequalities. Cambridge university press.
Dragomir, S. S., & Agarwal, R. (1998). Two inequalities for differentiable mappings and applications to special means of real numbers and to trapezoidal formula. Applied Mathematics Letters, 11(5), 91-95.
Kittaneh, F. (2005). Numerical radius inequalities for Hilbert space operators. Studia Mathematica, 168(1), 73-80.
