Hermite–Hadamard’s inequality and Cauchy-type mean operators for positive -semigroups

Abstract

Hermite–Hadamard’s-type inequality is derived using superquadratic maps for positive operator semigroups. A methodical procedure was adopted to obtain the corresponding mean operators.

Keywords:

• Hermite–Hadamard’s-type inequalities
• positive semigroups
• mean operators

Public Interest Statement

This research is devoted to Cauchy-type mean Operators defined on superquadratic mappings and operator semigroups. Gul I Hina Aslam and Matloob Anwar have been generalizing the theory of inequalities for operator semigroups. In this paper, authors continue the study for superquadratic mappings.

1. Preambles

The idea of the main content of this note is not unusual, but is this decade’s active research area (Bakry, 2004; Wang, 2005; Wang, 2013). In recent years, there has been considerable interest in the generalization of functional and type-inequalities to the operator semigroups. To get a common abstract setting within which ordering among the elements can be considered, a Banach lattice was defined. So that the idea related to positivity can be generalized.

Any (real) vector space $\mathfrak{E}$ is said to be an ordered vector space, if

$A_1$:	=	$\mathfrak{f}\le \mathfrak{g}$$\Rightarrow$$\mathfrak{f}+\mathfrak{h}\le \mathfrak{g}+\mathfrak{h}$ for all $\mathfrak{f},\mathfrak{g},\mathfrak{h}\in \mathfrak{E},$
$A_2$:	=	$\mathfrak{f}\ge 0$$\Rightarrow$$\alpha \mathfrak{f}\ge 0$$\forall$$\mathfrak{f}\in \mathfrak{E}$ and $\alpha \ge 0,$

Note that, $A_1$, shows the translation invariance. Therefore, it indicates that the order among the elements of $\mathfrak{E}$ is completely established by the positive part ${\mathfrak{E}}_{+}=\{\mathfrak{f}\in \mathfrak{E}:\mathfrak{f}\ge 0\}$ of $\mathfrak{E}$. Equivalently, $\mathfrak{f}\le \mathfrak{g}$ if and only if $\mathfrak{g}-\mathfrak{f}\in {\mathfrak{E}}_{+}$. Moreover, $A_2$ indicates that ${\mathfrak{E}}_{+}$ is a convex set and a cone with vertex 0. If a “supremum” $sup(\mathfrak{f},\mathfrak{g})$ and thus an “infumum” $inf(\mathfrak{f},\mathfrak{g})$ for any two elements $\mathfrak{f},\mathfrak{g}\in \mathfrak{E}$ can be specified, an ordered vector space $\mathfrak{E}$ is called a vector lattice. The compatibility axiom between norm and order is given briefly bellow:(1.1) $$\begin{array}{c}\hfill |\mathfrak{f}|\le |\mathfrak{g}|\Rightarrow \Vert \mathfrak{f}\Vert \le \Vert \mathfrak{g}\Vert ,\mathfrak{f},\mathfrak{g}\in \mathfrak{E},\end{array}$$(1.1)

where $sup(\mathfrak{f},-\mathfrak{f})=|\mathfrak{f}|$. This particular norm is called a lattice norm. Therefore, a vector lattice $(\mathfrak{E},\le )$ establishing a lattice norm is called Banach lattice. If a Banach lattice satisfies that, $\mathfrak{f},\mathfrak{g}\in {\mathfrak{E}}_{+}\Rightarrow \mathfrak{f}\mathfrak{g}\in {\mathfrak{E}}_{+}$, it is said to be a Banach lattice algebra. A linear mapping $\mathfrak{L}:\mathfrak{E}\to \mathfrak{E}$ is positive ($\mathfrak{L}\ge 0$) if $\mathfrak{L}(\mathfrak{f})\in {\mathfrak{E}}_{+}$, for all $\mathfrak{f}\in {\mathfrak{E}}_{+}$. The set of all positive linear mappings forms a convex cone in the space $L(\mathfrak{E})$ and defines the natural ordering in it. The absolute value of $\mathfrak{L}$ is given bellow (if exists),$$\begin{array}{c}\hfill |\mathfrak{L}|(\mathfrak{f})=sup\{\mathfrak{L}(\mathfrak{g}):|\mathfrak{g}|\le \mathfrak{f}\},(\mathfrak{f}\in {\mathfrak{E}}_{+}).\end{array}$$

Therefore, $\mathfrak{L}:\mathfrak{E}\to \mathfrak{E}$ is positive iff $|\mathfrak{L}(\mathfrak{f})|\le \mathfrak{L}(|\mathfrak{f}|)$ for any $\mathfrak{f}\in \mathfrak{E}$.

Definition 1.1

A family of bounded linear operators ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ on a Banach space $\mathfrak{E}$ is called a (one parameter) $C_0$-semigroup (or strongly continuous semigroup), if it satisfies

(i)	$\mathfrak{T}(s)\mathfrak{T}(t)=\mathfrak{T}(s+t)$ for all $s,t\in {\mathbb{R}}^{+}$.
(ii)	$\mathfrak{T}(0)=I$

and is strongly continuous in the sense that for every $\mathfrak{f}\in \mathfrak{E}$ the orbit maps$$\begin{array}{c}\hfill :{\zeta }_{\mathfrak{f}}:s\to {\zeta }_{\mathfrak{f}}(s):=T(s)\mathfrak{f}\end{array}$$

are continuous from ${\mathbb{R}}_{+}$ into $\mathfrak{E}$ for every $\mathfrak{f}\in \mathfrak{E}$.

The (infinitesimal) generator $A:\mathfrak{E}\supseteq D(A)\to R(A)\subseteq \mathfrak{E}$ of a strongly continuous semigroup is the densely defined closed linear operator$$\begin{array}{cc}\hfill A\mathfrak{f}& ={\dot{\zeta }}_{\mathfrak{f}}(0)=\underset{h\to 0^{+}}{lim}\frac{1}{s}(\mathfrak{T}(s)\mathfrak{f}-\mathfrak{f})(\mathfrak{f}\in D(A))\hfill \\ \hfill D(A)& =\{\mathfrak{f}:\underset{s\to 0^{+}}{lim}\frac{1}{s}(\mathfrak{T}(s)\mathfrak{f}-\mathfrak{f})\text{exists in}\mathfrak{E}.\}\hfill \end{array}$$

For a positive $C_0$-semigroup ${\{\mathfrak{T}(s)\}}_{s\ge 0}$, the positivity of the operators can be equivalent to:$$\begin{array}{c}\hfill |\mathfrak{T}(s)\mathfrak{f}|\le \mathfrak{T}(s)|\mathfrak{f}|,t\ge 0,\mathfrak{f}\in \mathfrak{E}.\end{array}$$

More details can be found in Nagel (1986).

In accordance with the customary definition of power integral means , the power means for $C_0$-group of operators are defined.

Definition 1.2

(Aslam & Anwar, 2015a) Let ${\{\mathfrak{T}(s)\}}_{s\in \mathbb{R}}$ be the $C_0$-group of operators on a Banach space $\mathcal{X}$. The power mean is given as(1.2) $$\begin{array}{c}\hfill {\mathcal{M}}_r(\mathfrak{T},\mathfrak{f},s)=\left\{\begin{array}{cc}\{\frac{1}{s}{\int }_0^s{[\mathfrak{T}(ϵ)]}^r\mathfrak{f}dϵ{\}}^{1/r},\hfill & r\ne 0\hfill \\ \hfill \\ exp[\frac{1}{s}{\int }_0^{s}ln[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ],\hfill & r=0.\hfill \end{array}\right.\end{array}$$(1.2)

where $\mathfrak{f}\in \mathcal{X}$ and $s\in \mathbb{R}$.

2. Main Results

In Banić and Varošanec , (2008), the Hermite–Hadamard’s-type inequality for positive linear functionals is derived. Few mean-value theorems that ultimately lead to new means of Cauchy type are given in Abramovich et al. , (2010).

In this note, we generalize the Hermite–Hadamard’s-type inequality for positive $C_0$-semigroup. We also give some generalized mean value theorems and define related mean operators.

Throughout the section, $\mathfrak{E}$ denotes the real Banach lattice algebra, until and unless stated otherwise.

Definition 2.1

(Aslam & Anwar, 2015b) An operator $\mathit{\psi }:{\mathfrak{E}}_{+}\to \mathfrak{E}$ is superquadratic, given that for all ${\mathfrak{g}}_1\ge 0$ there exists a constant vector $C({\mathfrak{g}}_1)$ such that(2.1) $$\begin{array}{c}\hfill \mathit{\psi }({\mathfrak{f}}_1)-\mathit{\psi }({\mathfrak{g}}_1)-\mathit{\psi }(|{\mathfrak{f}}_1-{\mathfrak{g}}_1|)\ge C({\mathfrak{g}}_1)({\mathfrak{f}}_1-{\mathfrak{g}}_1)\end{array}$$(2.1)

for all ${\mathfrak{f}}_1\ge 0$.

Theorem 2.2

Let ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ be a positive $C_0$-semigroup on $\mathfrak{E}$; then for an integrable superquadratic operator $\mathit{\psi }:{\mathfrak{E}}_{+}\to \mathfrak{E}$, we have(2.2) $$\begin{array}{c}\hfill \mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+\frac{1}{s}{\int }_0^s\psi [|[\mathfrak{T}(ϵ)]f-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ\le \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ,f\in {\mathfrak{E}}_{+}.\end{array}$$(2.2)

Proof

Let $\mathit{\psi }$ be a superquadratic mapping, then (2.1(2.1) $$\begin{array}{c}\hfill \mathit{\psi }({\mathfrak{f}}_1)-\mathit{\psi }({\mathfrak{g}}_1)-\mathit{\psi }(|{\mathfrak{f}}_1-{\mathfrak{g}}_1|)\ge C({\mathfrak{g}}_1)({\mathfrak{f}}_1-{\mathfrak{g}}_1)\end{array}$$(2.1) ) holds for all $\mathfrak{f},\mathfrak{g}\in {\mathfrak{E}}_{+}$. Choosing ${\mathfrak{f}}_1=[\mathfrak{T}(ϵ)]\mathfrak{f}$ and ${\mathfrak{g}}_1=\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ$ in (2.1(2.1) $$\begin{array}{c}\hfill \mathit{\psi }({\mathfrak{f}}_1)-\mathit{\psi }({\mathfrak{g}}_1)-\mathit{\psi }(|{\mathfrak{f}}_1-{\mathfrak{g}}_1|)\ge C({\mathfrak{g}}_1)({\mathfrak{f}}_1-{\mathfrak{g}}_1)\end{array}$$(2.1) ) we get$$\begin{array}{ccc}\hfill \mathit{\psi }[[\mathfrak{T}(ϵ)]\mathfrak{f}]& \ge \hfill & \hfill \mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+C[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ][[\mathfrak{T}(ϵ)]\mathfrak{f}-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}d\tau ]\\ \hfill & +\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-\frac{1}{t}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]\hfill \end{array}$$

By integrating from $0\to s$, we obtain,$$\begin{array}{ccc}\hfill {\int }_0^s\mathit{\psi }[[\mathfrak{T}(ϵ)]\mathfrak{f}]dϵ& \ge \hfill & \hfill t.\mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+C[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ][{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ-t\{\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ\}]\\ \hfill & +{\int }_0^s\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ,\hfill \end{array}$$

or$$\begin{array}{c}\hfill {\int }_0^s\mathit{\psi }[[\mathfrak{T}(ϵ)]\mathfrak{f}]dϵ\ge s.\mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+{\int }_0^s\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ.\end{array}$$

By multiplying 1 / s, we finally get the assertion (2.2(2.2) $$\begin{array}{c}\hfill \mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+\frac{1}{s}{\int }_0^s\psi [|[\mathfrak{T}(ϵ)]f-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ\le \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ,f\in {\mathfrak{E}}_{+}.\end{array}$$(2.2) ).

Definition 2.3

Let ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ be a positive $C_0$-semigroup of operators defined on $\mathfrak{E}$; then for an integrable operator $\mathit{\psi }:{\mathfrak{E}}_{+}\to \mathfrak{E}$, we define another operator ${\mathrm{\Lambda }}_{\mathit{\psi }}:{\mathfrak{E}}_{+}\to \mathfrak{E}$(2.3) $$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\psi }:=\frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ-\mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]-\frac{1}{s}{\int }_0^s\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ,\mathfrak{f}\in {\mathfrak{E}}_{+}.\end{array}$$(2.3)

If $\mathit{\psi }$ is continuous superquadratic mapping, then by (2.2(2.2) $$\begin{array}{c}\hfill \mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+\frac{1}{s}{\int }_0^s\psi [|[\mathfrak{T}(ϵ)]f-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ\le \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ,f\in {\mathfrak{E}}_{+}.\end{array}$$(2.2) ), ${\mathrm{\Lambda }}_{\mathit{\psi }}\ge 0$.

$\square$

To simplify expressions, we replace $\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ$ by ${\mathcal{M}}_1(s)$. Therefore, ${\mathrm{\Lambda }}_{\mathit{\psi }}$ can be written as$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\mathit{\psi }}:=\frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ-\mathit{\psi }[{\mathcal{M}}_1(s)]-\frac{1}{s}{\int }_0^s\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-{\mathcal{M}}_1(s)|]d\tau \end{array}$$

The operator analogue of Abramovich et al. (2004, Lemma 3.1) is given in Aslam and Anwar (2015b).

Lemma 2.4

Let $\mathit{\psi }:{\mathfrak{E}}_{+}\to \mathfrak{E}$ be continuously differentiable and $\mathit{\psi }(0)\le 0$. If ${\mathit{\psi }}^{\prime }$ is super-additive or $\mathfrak{f}\to \frac{{\mathit{\psi }}^{\prime }(\mathfrak{g})}{\mathfrak{g}},\mathfrak{g}\in {\mathfrak{E}}_{+}$, is increasing, then $\mathit{\psi }$ is superquadratic.

$\square$

Lemma 2.5

Let $\mathit{\psi }\in C^2[{\mathfrak{E}}_{+}]$ and $u,U\in \mathfrak{E}$ be such that(2.4) $$\begin{array}{c}\hfill u\le (\frac{{\mathit{\psi }}^{\prime }(\mathfrak{g})}{\mathfrak{g}}{)}^{\prime }=\frac{\mathfrak{f}{\mathit{\psi }}^{″}(\mathfrak{g})-{\mathit{\psi }}^{\prime }(\mathfrak{g})}{{\mathfrak{g}}^2}\le U,\text{for all}\mathfrak{g}>0.\end{array}$$(2.4)

Consider the operators ${\mathit{\psi }}_1,{\mathit{\psi }}_2:{\mathfrak{E}}_{+}\to \mathfrak{E}$ defined as:$$\begin{array}{c}\hfill {\mathit{\psi }}_1(\mathfrak{g})=\frac{U{\mathfrak{g}}^3}{3}-\mathit{\psi }(\mathfrak{g}),{\mathit{\psi }}_2=\mathit{\psi }(\mathfrak{g})-\frac{u{\mathfrak{g}}^3}{3}\end{array}$$

Then, the mappings $\mathfrak{g}\to \frac{{\mathit{\psi }}_1^{\prime }(\mathfrak{g})}{\mathfrak{g}}$ and $\mathfrak{g}\to \frac{{\mathit{\psi }}_2^{\prime }(\mathfrak{g})}{\mathfrak{g}}$ are increasing. If also ${\mathit{\psi }}_i(0)=0,i=1,2$, then these are superquadratic mappings.

Proof

Using (2.4(2.4) $$\begin{array}{c}\hfill u\le (\frac{{\mathit{\psi }}^{\prime }(\mathfrak{g})}{\mathfrak{g}}{)}^{\prime }=\frac{\mathfrak{f}{\mathit{\psi }}^{″}(\mathfrak{g})-{\mathit{\psi }}^{\prime }(\mathfrak{g})}{{\mathfrak{g}}^2}\le U,\text{for all}\mathfrak{g}>0.\end{array}$$(2.4) ), it can be noted that the mappings $\mathfrak{g}\to \frac{{\mathit{\psi }}_1^{\prime }(\mathfrak{g})}{\mathfrak{g}}$ and $\mathfrak{g}\to \frac{{\mathit{\psi }}_2^{\prime }(\mathfrak{g})}{\mathfrak{g}}$ are increasing. Moreover, if ${\mathit{\psi }}_i(0)=0,i=1,2$, Lemma (2.4) implies these to be superquadratic. $\square$

Theorem 2.6

Let ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ be a positive $C_0$-semigroup on $\mathfrak{E}$ and $\frac{{\mathit{\psi }}^{\prime }}{\mathfrak{f}}\in C^1({\mathfrak{E}}_{+})$ and $\mathit{\psi }(0)=0$, then the following inequality holds(2.5) $$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\mathit{\psi }}=\frac{\rho {\psi }^{″}(\rho )-{\mathit{\psi }}^{\prime }(\rho )}{3{\rho }^2}\{{({\mathcal{M}}_3(s))}^3-{({\mathcal{M}}_1(s))}^3-\frac{1}{s}{\int }_0^s|\mathfrak{T}(ϵ)\mathfrak{f}-m_s{|}^{3}dϵ\}\end{array}$$(2.5)

Here, $m_s$ denotes the arithmetic mean of ${\{\mathfrak{T}(s)\}}_{s\ge 0}$.

Proof

Suppose the conditions in Lemma 2.5 hold for all $\mathfrak{f}\in {\mathfrak{E}}_{+}$. Using ${\mathit{\psi }}_1$ instead of $\mathit{\psi }$ in (2.2(2.2) $$\begin{array}{c}\hfill \mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+\frac{1}{s}{\int }_0^s\psi [|[\mathfrak{T}(ϵ)]f-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ\le \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ,f\in {\mathfrak{E}}_{+}.\end{array}$$(2.2) ), we get$$\begin{array}{cc}\hfill \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ-\mathit{\psi }[{\mathcal{M}}_1(s)]-\frac{1}{s}{\int }_0^s\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-{\mathcal{M}}_1(s)|]dϵ& \le \frac{U}{3}\{{({\mathcal{M}}_3(s))}^3-{({\mathcal{M}}_1(s))}^3\hfill \\ \hfill & -\frac{1}{s}{\int }_0^s|\mathfrak{T}(ϵ)\mathfrak{f}-m_s{|}^{3}dϵ\}\hfill \end{array}$$

Similarly, using ${\mathit{\psi }}_2$ instead of $\mathit{\psi }$ in (2.2(2.2) $$\begin{array}{c}\hfill \mathit{\psi }[\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ]+\frac{1}{s}{\int }_0^s\psi [|[\mathfrak{T}(ϵ)]f-\frac{1}{s}{\int }_0^s[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ|]dϵ\le \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ,f\in {\mathfrak{E}}_{+}.\end{array}$$(2.2) ), we get$$\begin{array}{ccc}\hfill \frac{1}{s}{\int }_0^s\mathit{\psi }[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ-\mathit{\psi }[{\mathcal{M}}_1(s)]-\frac{1}{s}{\int }_0^s\mathit{\psi }[|[\mathfrak{T}(ϵ)]\mathfrak{f}-{\mathcal{M}}_1(s)|]dϵ& \ge \hfill & \hfill \frac{u}{3}\{{({\mathcal{M}}_3(s))}^3-{({\mathcal{M}}_1(s))}^3\\ \hfill & -\frac{1}{s}{\int }_0^s|\mathfrak{T}(ϵ)\mathfrak{f}-m_s{|}^{3}dϵ\}\hfill \end{array}$$

By combining the above two inequalities and using intermediate value theorem Ali (1997), we have existence of $\rho \in {\mathfrak{E}}_{+}$ such that (2.5(2.5) $$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\mathit{\psi }}=\frac{\rho {\psi }^{″}(\rho )-{\mathit{\psi }}^{\prime }(\rho )}{3{\rho }^2}\{{({\mathcal{M}}_3(s))}^3-{({\mathcal{M}}_1(s))}^3-\frac{1}{s}{\int }_0^s|\mathfrak{T}(ϵ)\mathfrak{f}-m_s{|}^{3}dϵ\}\end{array}$$(2.5) ) holds. $\square$

Theorem 2.7

Let ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ be a positive $C_0$-semigroup on $\mathfrak{E}$ and $\frac{{\mathit{\psi }}^{\prime }}{\mathfrak{f}},\frac{{\mathit{\varphi }}^{\prime }}{\mathfrak{f}}\in C^1({\mathfrak{E}}_{+})$ such that, $\mathit{\psi }(0)=\mathit{\varphi }(0)=0$, we have(2.6) $$\begin{array}{c}\hfill \frac{{\mathrm{\Lambda }}_{\mathit{\psi }}}{{\mathrm{\Lambda }}_{\mathit{\varphi }}}=\frac{\rho {\mathit{\psi }}^{″}(\rho )-{\mathit{\psi }}^{\prime }(\rho )}{\xi {\varphi }^{″}(\rho )-{\rho }^{\prime }(\rho )}=\mathbf{F}(\rho ),\rho \in {\mathfrak{E}}_{+},\end{array}$$(2.6)

given the denominator is not zero. If ${\mathbf{F}}^{-1}$ exists, then(2.7) $$\begin{array}{c}\hfill \rho ={\mathbf{F}}^{-1}(\frac{{\mathrm{\Lambda }}_{\mathit{\psi }}}{{\mathrm{\Lambda }}_{\varphi }}),{\mathrm{\Lambda }}_{\varphi }\ne 0,\end{array}$$(2.7)

Proof

Consider a function $\mathrm{\Omega }=c_1\mathit{\psi }-c_2\varphi$, where$$\begin{array}{c}\hfill c_1={\mathrm{\Lambda }}_{\varphi },c_2={\mathrm{\Lambda }}_{\mathit{\psi }}.\end{array}$$

Then, for $\mathfrak{f}\in {\mathfrak{E}}_{+}$$$\begin{array}{c}\hfill \frac{{\mathrm{\Omega }}^{\prime }}{\mathfrak{f}}=c_1\frac{{\mathit{\psi }}^{\prime }}{\mathfrak{f}}-c_2\frac{{\varphi }^{\prime }}{\mathfrak{f}}\in C^1({\mathfrak{E}}_{+}).\end{array}$$

we can calculate that ${\mathrm{\Lambda }}_{\mathrm{\Omega }}=0$ and by Lemma (2.5) with $\mathit{\psi }=\mathrm{\Omega }$ we have$$\begin{array}{c}\hfill [c_1(\rho {\mathit{\psi }}^{″}(\rho )-{\mathit{\psi }}^{\prime }(\rho ))-c_2(\rho {\varphi }^{″}(\rho )-{\varphi }^{\prime }(\rho ))][{({\mathcal{M}}_3(t))}^3-{({\mathcal{M}}_1(t))}^3-\frac{1}{s}{\int }_0^s{|\mathfrak{T}(\tau )\mathfrak{f}-m_t|}^{3}d\tau ]=0,\mathfrak{f}\in {\mathfrak{E}}_{+}.\end{array}$$

Since $\mathit{\psi }={\mathfrak{f}}^3$ is superquadratic mapping and ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ is positive, we have$$\begin{array}{c}\hfill \frac{c_2}{c_1}=\frac{\rho {\mathit{\psi }}^{″}(\rho )-{\mathit{\psi }}^{\prime }(\rho )}{\rho {\varphi }^{″}(\rho )-{\varphi }^{\prime }(\rho )}=\frac{{\mathrm{\Lambda }}_{\mathit{\psi }}}{{\mathrm{\Lambda }}_{\varphi }},\rho \in {\mathfrak{E}}_{+},\end{array}$$

given the denominator is not zero, proof follows. $\square$

Let $\mathcal{G}$ be the set of all invertible bounded linear operators $H:\mathfrak{E}\to \mathfrak{E}$. For a $C_0$-semigroup of positive operators ${\{\mathfrak{T}(s)\}}_{s\ge 0}\subset B(\mathfrak{E})$ defined on $\mathfrak{E}$ and $H\in \mathcal{G}$, the quasi-arithmetic mean is given as Aslam and Anwar (2015a)(2.8) $$\begin{array}{c}\hfill {\mathcal{M}}_H^{\circ }(\mathfrak{T},\mathfrak{f},s)=H^{-1}\{\frac{1}{s}{\int }_0^{s}H[\mathfrak{T}(ϵ)\mathfrak{f}]dϵ\},\mathfrak{f}\in {\mathfrak{E}}_{+},s\ge 0.\end{array}$$(2.8)

By Belleni-Morante and McBride (1998, Lemma 1.85), $B(\mathfrak{E})$ is closed under composition of operators, therefore the above expressions exist and ${\mathcal{M}}_H^{\circ }(\mathfrak{T},\mathfrak{f},s)\in \mathfrak{E}$. To avoid complexity among the expressions, let$$\begin{array}{c}\hfill C^2\mathcal{G}(\mathfrak{E})=\{H:H\in \mathcal{G},H^{″}\text{exists in Gateaux sense}\}\end{array}$$.

Theorem 2.8

Let ${\{\mathfrak{T}(s)\}}_{s\ge 0}$ be a positive $C_0$-semigroup of operators defined on $\mathfrak{E}$ and $\mathbf{H},\mathbf{F},\mathbf{K}\in C^2\mathcal{G}(\mathfrak{E})$. Let for $\mathfrak{f}\in {\mathfrak{E}}_{+}$,$\frac{\mathbf{H}\circ {\mathbf{F}}^{-1}(\mathfrak{f})}{\mathfrak{f}},\frac{\mathbf{K}\circ {\mathbf{F}}^{-1}(\mathfrak{f})}{\mathfrak{f}}\in C^1(\mathfrak{E})$ with $\mathbf{H}\circ {\mathbf{F}}^{-1}(0)=0=\mathbf{K}\circ {\mathbf{F}}^{-1}(0)$, then for $\mathfrak{f}\in {\mathfrak{E}}_{+}$(2.9) $$\begin{array}{cc}& \frac{\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}{\mathbf{K}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{K}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}\hfill \\ \hfill & =\frac{\mathbf{F}(\xi )\{{\mathbf{H}}^{″}(\xi ){\mathbf{F}}^{\prime }(\xi )-{\mathbf{H}}^{\prime }(\xi ){\mathbf{F}}^{″}(\xi )-{\mathbf{H}}^{\prime }(\xi ){[{\mathbf{F}}^{\prime }(\xi )]}^2\}}{\mathbf{F}(\xi )\{{\mathbf{K}}^{″}(\xi ){\mathbf{F}}^{\prime }(\xi )-{\mathbf{K}}^{\prime }(\xi ){\mathbf{F}}^{″}(\xi )-{\mathbf{K}}^{\prime }(\xi ){[{\mathbf{F}}^{\prime }(\xi )]}^2\}},\hfill \end{array}$$(2.9)

holds for some $\xi \in {\mathfrak{E}}_{+}$, provided the denominator does not vanish.

Proof

By setting the operators $\mathit{\psi }$ and $\varphi$ in Theorem 2.7, such that$$\begin{array}{c}\hfill \mathit{\psi }=\mathbf{H}\circ {\mathbf{F}}^{-1},\varphi =\mathbf{K}\circ {\mathbf{F}}^{-1}and\mathfrak{T}(ϵ)\mathfrak{f}=\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}],\mathfrak{f}\in {\mathfrak{E}}_{+},\end{array}$$

where $\mathbf{H},\mathbf{F},\mathbf{K}\in C^2\mathcal{G}(\mathfrak{E})$. We find that there exists $\rho \in {\mathfrak{E}}_{+}$, such that$$\begin{array}{cc}& \frac{\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}{\mathbf{K}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{K}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}\hfill \\ \hfill & =\frac{\rho \{{\mathbf{H}}^{″}({\mathbf{F}}^{-1}\rho ){\mathbf{F}}^{\prime }({\mathbf{F}}^{-1}\rho )-{\mathbf{H}}^{\prime }({\mathbf{F}}^{-1}\rho ){\mathbf{F}}^{″}({\mathbf{F}}^{-1}\rho )-{\mathbf{H}}^{\prime }({\mathbf{F}}^{-1}\rho ){[{\mathbf{F}}^{\prime }({\mathbf{F}}^{-1}\rho )]}^2\}}{\rho \{{\mathbf{K}}^{″}({\mathbf{F}}^{-1}\rho ){\mathbf{F}}^{\prime }({\mathbf{F}}^{-1}\rho )-{\mathbf{K}}^{\prime }({\mathbf{F}}^{-1}\rho ){\mathbf{F}}^{″}({\mathbf{F}}^{-1}\rho )-{\mathbf{K}}^{\prime }({\mathbf{F}}^{-1}\rho ){[{\mathbf{F}}^{\prime }({\mathbf{F}}^{-1}\rho )]}^2\}},\hfill \end{array}$$

Hence, by putting ${\mathbf{F}}^{-1}(\rho )=\omega$ for some $\mu \in \mathfrak{E}$, the assertion (2.9(2.9) $$\begin{array}{cc}& \frac{\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}{\mathbf{K}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{K}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}\hfill \\ \hfill & =\frac{\mathbf{F}(\xi )\{{\mathbf{H}}^{″}(\xi ){\mathbf{F}}^{\prime }(\xi )-{\mathbf{H}}^{\prime }(\xi ){\mathbf{F}}^{″}(\xi )-{\mathbf{H}}^{\prime }(\xi ){[{\mathbf{F}}^{\prime }(\xi )]}^2\}}{\mathbf{F}(\xi )\{{\mathbf{K}}^{″}(\xi ){\mathbf{F}}^{\prime }(\xi )-{\mathbf{K}}^{\prime }(\xi ){\mathbf{F}}^{″}(\xi )-{\mathbf{K}}^{\prime }(\xi ){[{\mathbf{F}}^{\prime }(\xi )]}^2\}},\hfill \end{array}$$(2.9) ) follows directly. $\square$

The above theorem enables us to introduce new means. Set$$\begin{array}{c}\hfill \mathrm{\Gamma }(\omega )=\frac{\mathbf{F}(\omega )\{{\mathbf{H}}^{″}(\omega ){\mathbf{F}}^{\prime }(\omega )-{\mathbf{H}}^{\prime }(\omega ){\mathbf{F}}^{″}(\omega )-{\mathbf{H}}^{\prime }(\omega ){[{\mathbf{F}}^{\prime }(\omega )]}^2\}}{\mathbf{F}(\omega )\{{\mathbf{K}}^{″}(\omega ){\mathbf{F}}^{\prime }(\omega )-{\mathbf{K}}^{\prime }(\omega ){\mathbf{F}}^{″}(\omega )-{\mathbf{K}}^{\prime }(\omega ){[{\mathbf{F}}^{\prime }(\omega )]}^2\}},\end{array}$$

and when $\mathbf{F}\in \mathcal{G}(\mathfrak{E})$$$\begin{array}{c}\hfill \omega ={\mathrm{\Gamma }}^{-1}(\frac{\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }({\mathit{\psi }}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}{\mathbf{K}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{K}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))})\end{array}$$

Remark 2.9

For a Banach lattice algebra $(\mathfrak{E},\Vert .\Vert )$, from Theorem 2.8 we have that$$\begin{array}{c}\hfill m\le \Vert \frac{\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{H}({\mathcal{M}}_{\mathbf{H}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}{\mathbf{K}({\mathcal{M}}_{\mathbf{H}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s))-\mathbf{K}({\mathcal{M}}_{\mathbf{K}}^{\circ }({\mathbf{F}}^{-1}|\mathbf{F}[\mathfrak{T}(ϵ)\mathfrak{f}]-\mathbf{F}{\mathcal{M}}_{\mathbf{F}}^{\circ }(\mathfrak{T},\mathfrak{f},s)|,\mathfrak{f},s))}\Vert \le M,\end{array}$$

where m and M are, respectively, the minimum and maximum values of$$\begin{array}{c}\hfill \Vert \frac{\mathbf{F}(\omega )\{{\mathbf{H}}^{″}(\omega ){\mathbf{F}}^{\prime }(\omega )-{\mathbf{H}}^{\prime }(\omega ){\mathbf{F}}^{″}(\omega )-{\mathbf{H}}^{\prime }(\omega ){[{\mathbf{F}}^{\prime }(\omega )]}^2\}}{\mathbf{F}(\omega )\{{\mathbf{K}}^{″}(\omega ){\mathbf{F}}^{\prime }(\omega )-{\mathbf{K}}^{\prime }(\omega ){\mathbf{F}}^{″}(\omega )-{\mathbf{K}}^{\prime }(\omega ){[{\mathbf{F}}^{\prime }(\omega )]}^2\}}\Vert ,\omega \in \mathfrak{E}.\end{array}$$

$\square$

Next we give a significant result which leads us to define the Cauchy-type mean operators on $C_0$-group of operators.

Corollary 2.10

Let all the conditions of Theorem 2.8 be satisfied. For $r,n,l\in {\mathbb{R}}_{+}$ such that $r\ne l;l\ne 2s$, we have(2.10) $$\begin{array}{c}\hfill \frac{{\mathcal{M}}_r^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_r^r{(|[\mathfrak{T}(\tau )\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}{{\mathcal{M}}_l^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_l^l{(|[\mathfrak{T}(ϵ)\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}=\frac{r(r-2n)}{l(l-2n)}{\omega }^{r-l}\end{array}$$(2.10)

where ${\mathcal{M}}_r(\mathfrak{T},\mathfrak{f},s)$ is defined by (1.2(1.2) $$\begin{array}{c}\hfill {\mathcal{M}}_r(\mathfrak{T},\mathfrak{f},s)=\left\{\begin{array}{cc}\{\frac{1}{s}{\int }_0^s{[\mathfrak{T}(ϵ)]}^r\mathfrak{f}dϵ{\}}^{1/r},\hfill & r\ne 0\hfill \\ \hfill \\ exp[\frac{1}{s}{\int }_0^{s}ln[\mathfrak{T}(ϵ)]\mathfrak{f}dϵ],\hfill & r=0.\hfill \end{array}\right.\end{array}$$(1.2) ). The assertion (2.10(2.10) $$\begin{array}{c}\hfill \frac{{\mathcal{M}}_r^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_r^r{(|[\mathfrak{T}(\tau )\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}{{\mathcal{M}}_l^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_l^l{(|[\mathfrak{T}(ϵ)\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}=\frac{r(r-2n)}{l(l-2n)}{\omega }^{r-l}\end{array}$$(2.10) ) holds for some $\mu$, provided that the denominators do not vanish.

Proof

For $r,n,l\in {\mathbb{R}}_{+}$ and $\mathfrak{f}\in {\mathfrak{E}}_{+}$, if we set$$\begin{array}{c}\hfill \mathbf{H}(\mathfrak{f})={\mathfrak{f}}^r,\mathbf{F}(\mathfrak{f})={\mathfrak{f}}^n,\mathbf{K}(\mathfrak{f})={\mathfrak{f}}^l\end{array}$$

in Theorem (2.8), the assertion in (2.10(2.10) $$\begin{array}{c}\hfill \frac{{\mathcal{M}}_r^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_r^r{(|[\mathfrak{T}(\tau )\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}{{\mathcal{M}}_l^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_l^l{(|[\mathfrak{T}(ϵ)\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}=\frac{r(r-2n)}{l(l-2n)}{\omega }^{r-l}\end{array}$$(2.10) ) follows directly.

Finally, we are able to define mean operators of the Cauchy type on positive $C_0$-semigroup on Banach lattice algebra $\mathfrak{E}$.

Definition 2.11

Let $r,n,l\in {\mathbb{R}}_{+}$ and ${\{\mathfrak{T}(s)\}}_{s\ge 0}\subset B(\mathfrak{E})$ be a positive $C_0$-semigroup on a Banach lattice algebra $\mathfrak{E}$. Then,(2.11) $$\begin{array}{c}\hfill {\mathfrak{M}}_r^{l,n}(\mathfrak{T},\mathfrak{f},s)=(\frac{l(l-2n)}{r(r-2n)}\frac{{\mathcal{M}}_r^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^r(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_r^r{(|[\mathfrak{T}(ϵ)\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}{{\mathcal{M}}_l^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_n^l(\mathfrak{T},\mathfrak{f},s)-{\mathcal{M}}_l^l{(|[\mathfrak{T}(ϵ)\mathfrak{f}]}^n-{\mathcal{M}}_n^n(\mathfrak{T},\mathfrak{f},s){|}^{\frac{1}{s}},\mathfrak{f},s)}{)}^{\frac{1}{r-l}}.\end{array}$$(2.11)

is a family of mean operators of the Cauchy type on $C_0$-semigroup of positive operators. This definition is true for all $r\ne l\ne n\ne 0$ and other cases can be taken as limiting cases.

3. Conclusion

In this note, a Hermite-Hadamard type inequality has been proved for a positive $C_0$-semigroup and a superquadratic mapping defined on a Banach lattice algebra. A methodic way has been adopted to prove the corresponding mean value theorems, which enabled us to define a new set of mean operators. These mean operators are inspired from Cauchy-type mean.

Additional information

Funding

The authors received no direct funding for this research.

Notes on contributors

Gul I. Hina Aslam

Gul I Hina Aslam is an assistant professor of Mathematics in the Department of Applied Sciences, Pakistan Navy Engineering College, National University of Sciences and Technology, Karachi, Pakistan. Her area of interest is Operator semi-groups, Inequalities and Applied Analysis.

Matloob Anwar

Gul I Hina Aslam is an assistant professor of Mathematics in the Department of Applied Sciences, Pakistan Navy Engineering College, National University of Sciences and Technology, Karachi, Pakistan. Her area of interest is Operator semi-groups, Inequalities and Applied Analysis.