1 Motivation

We begin our presentation with notation and terminology quoted from [6].

Let G be a compact connected Lie group and \( {\mathfrak {g}} \) be its Lie algebra. Assume T is a maximal torus of G and \( {\mathfrak {t}} \) is the Lie algebra of T. By \( {\mathfrak {t}}_+ \), we denote a closed Weyl chamber in \( {\mathfrak {t}} \). For a given element \( x \in {\mathfrak {g}} \), the symbol \( {\mathfrak {t}}_+ (x) \) represents the unique element of the set \( {\mathfrak {t}}_+ \cap G x \), where \( G x = \{ g \cdot x : g \in G \} \) and \( g \cdot x = ( \mathrm{Ad}\,g ) x \).

Let \( \langle \cdot , \cdot \rangle \) be a G-invariant inner product on \( {\mathfrak {g}} \). The dual cone of \( {\mathfrak {t}}_+ \) is given by \( {\mathfrak {t}}_+^*= \{ x \in {\mathfrak {g}} : \langle x , v \rangle \ge 0 \, \text{ for } \text{ all }\, v \in {\mathfrak {t}}_+ \} \). This cone generates the preorder \( \le \) on \( {\mathfrak {g}} \) by \( y \le x \; \text{ iff } \; x - y \in {\mathfrak {t}}_+^*\) for \( x , y \in {\mathfrak {g}} \). In addition, a related preorder \( \prec \) can be defined by \( y \prec x \; \text{ iff } \; y \in \mathrm{conv}\,G x \) for \( x , y \in {\mathfrak {g}} \), where \( \mathrm{conv}\,G x \) is the convex hull of the G-orbit \( \{ g \cdot x : g \in G \} \) (cf. [3, Corollary B.3]). It is known that \( \le \) and \( \prec \) coincide on \( {\mathfrak {t}}_+ \) [1, Proposition 18].

In [6, Theorem 7] T.-Y. Tam presented a triangle-type inequality for connected compact groups, as follows

$$\begin{aligned} {\mathfrak {t}}_+ ( x + y ) \le {\mathfrak {t}}_+ (x) + {\mathfrak {t}}_+ (y) \quad \text{ for }\,\, x ,y \in {\mathfrak {g}} , \end{aligned}$$
(1)

where \( {\mathfrak {t}}_+ ( z ) \) denotes the unique element in \( {\mathfrak {t}}_+ \cap G z \) corresponding to an element \( z \in {\mathfrak {g}} \).

A similar framework works in the context of real semisimple Lie algebras. Let \( {\mathfrak {g}} \) be a real semisimple Lie algebra with a Cartan decomposition \( {\mathfrak {g}} = {\mathfrak {k}} + {\mathfrak {p}} \), where \( {\mathfrak {p}} \ne 0 \). Let \( {{\mathfrak {a}}} \) be a maximal abelian subalgebra in \( {\mathfrak {p}} \) and \( {{\mathfrak {a}}}_+ \) be a closed Weyl chamber in \( {{\mathfrak {a}}} \).

The Killing form \( \langle \cdot , \cdot \rangle \) is positive definite on \( {\mathfrak {p}} \), so we can define the dual cone of \( {{\mathfrak {a}}}_+ \) by \( {{\mathfrak {a}}}_+^*= \{ x \in {\mathfrak {p}} : \langle x , v \rangle \ge 0 \; \text{ for } \text{ all } v \in {\mathfrak {p}} \}\). Then we introduce the preorder \( \le \) on \( {\mathfrak {p}} \) by \( y \le x \; \text{ iff } \; x - y \in {{\mathfrak {a}}}_+^*\) for \( x , y \in {\mathfrak {p}} \).

Let K be the maximal compact subgroup in the adjoint group \( \mathrm{Int}\,( {\mathfrak {g}} )\). So, \( \mathrm{Ad}\,K \) is maximal compact subgroup of \( \mathrm{Ad}\,G \). We define preorder \( \prec \) in the following manner: \( y \prec x \; \text{ iff } \; y \in \mathrm{conv}\,K x \) for \( x , y \in {\mathfrak {p}} \), where \( \mathrm{conv}\,K x \) is the convex hull of the K-orbit \( K x = \{ k \cdot x : k \in K \} \) with \( k \cdot x = ( \mathrm{Ad}\,k) x \) (cf. [3, Corollary B.3]). The preorders \( \le \) and \( \prec \) coincide on \( {{\mathfrak {a}}}_+ \) [2, Lemma 3.2]. For an element \( x \in {\mathfrak {p}} \), the symbol \( {{\mathfrak {a}}}_+ (x) \) represents the unique element of the set \( {{\mathfrak {a}}}_+ \cap K x \), where \( K x = \{ k \cdot x : k \in K \} \) and \( k \cdot x = ( \mathrm{Ad}\,k ) x \).

In [6, Theorem 2] T.-Y. Tam showed that

$$\begin{aligned} {{\mathfrak {a}}}_+ ( x + y ) \le {{\mathfrak {a}}}_+ (x) + {{\mathfrak {a}}}_+ (y) \quad \text{ for } x ,y \in {\mathfrak {p}}. \end{aligned}$$
(2)

In this note, our purpose is to show refinements of inequalities (1) and (2) by employing gradients of differentiable real functions increasing with respect to corresponding preorder \( \prec \) (cf. [6]).

2 Results for Compact Connected Lie Groups

In this section, we consider the Lie algebra \( {\mathfrak {g}} \) of a compact connected Lie group G. We use the norm \( \Vert {\cdot }\Vert = \langle \cdot , \cdot \rangle ^{1/2} \) on \( {\mathfrak {g}} \) generated by a G-invariant inner product \( \langle \cdot , \cdot \rangle \) on \( {\mathfrak {g}} \). The group G acts on \( {\mathfrak {g}} \) by \( \mathrm{Ad}\,G \). The preorder \( \prec \) is defined in Sect. 1.

A function \( \varPhi \) defined on \( {\mathfrak {g}} \) is called G-invariant if

$$\begin{aligned} \varPhi (g \cdot x) = \varPhi (x) \quad \text{ for } \text{ all } x \in {\mathfrak {g}} \quad \hbox {and} \quad g \in G. \end{aligned}$$

We say that a function \( \varPhi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) is G-increasing, if for \( x , y \in {\mathfrak {g}} \),

$$\begin{aligned} y \prec x \;\; \text{ implies } \;\; \varPhi (y) \le \varPhi (x). \end{aligned}$$

In order to state our results, we need to employ the convex cone \( {\mathcal C} \) of all G-increasing real functions defined on \( {\mathfrak {g}} \) as well as the cone preorder \( \le _{\mathcal C} \) generated by \( {\mathcal C} \), as follows: given two real functions \( \varphi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) and \( \psi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\), we use the notation \( \psi \le _{\mathcal C} \varphi \) whenever the difference function \( \varphi - \psi \) is G-increasing on \( {\mathfrak {g}} \).

Given two vectors \( x , y \in {\mathfrak {g}} \), if there exists a \( g \in G \) such that \( x \in g \, {\mathfrak {t}}_+ \) and \( y \in g \, {\mathfrak {t}}_+ \), then

$$\begin{aligned} {\mathfrak {t}}_+ ( x + y ) = {\mathfrak {t}}_+ ( x ) + {\mathfrak {t}}_+ ( y ). \end{aligned}$$
(3)

In this note, we take the convention that the Gateaux differentiability of a function \( \varPhi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) means the existence of the directional derivative

$$\begin{aligned} \nabla _h \varPhi (y) = \lim \limits _{t \rightarrow 0} \frac{ \varPhi (y + t h ) - \varPhi (y) }{t} \end{aligned}$$
(4)

at each point \( y \in {\mathfrak {g}} \) and in each direction \( h \in {\mathfrak {g}} \), and moreover that the map \( {\mathfrak {g}} \ni h \rightarrow \nabla _h \varPhi (y) \in {\mathbb {R}}\) is continuous and linear as a function of h. Consequently, there exists the gradient \( \nabla \varPhi (y) \in {\mathfrak {g}} \) satisfying the condition

$$\begin{aligned} \nabla _h \varPhi (y) = \langle \nabla \varPhi (y) , h \rangle \quad \text{ for } \text{ all }\quad h \in {\mathfrak {g}}. \end{aligned}$$
(5)

In accordance with [4, Theorem 2.1], a Gateaux differentiable G-increasing function \( \varPhi : {\mathfrak {g}} \rightarrow {\mathbb {R}}\) with continuous gradient \( \nabla \varPhi (\cdot ) \) satisfies the condition

$$\begin{aligned} \nabla \varPhi (g \, {\mathfrak {t}}_+ ( y ) ) \in g \, {\mathfrak {t}}_+ \;\;\; \text{ for } \text{ all } g \in G \text{ and } y \in {\mathfrak {g}} \text{. } \end{aligned}$$
(6)

Theorem 1

Let \( \varphi \) and \( \psi \) be Gateaux differentiable real functions on \( {\mathfrak {g}} \) with continuous gradients \( \nabla \varphi (\cdot )\) and \( \nabla \psi (\cdot )\), respectively. Suppose that

$$\begin{aligned} 0 \le _{\mathcal C} \psi \le _{\mathcal C} \varphi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 , \end{aligned}$$
(7)

i.e., the functions \( \psi \), \( \varphi - \psi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are G-increasing on \( {\mathfrak {g}} \).

If \( x , y \in {\mathfrak {g}} \) then

$$\begin{aligned} {\mathfrak {t}}_+ ( x + \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ) \prec {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \psi (y) ). \end{aligned}$$
(8)

Remark 1

Observe that for any points \( x , y \in {\mathfrak {g}} \), statement (8) in Theorem 1 shows the anti-isotonity of the functional

$$\begin{aligned} \phi \rightarrow {\mathfrak {t}}_+ ( x + \nabla \phi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \phi (y) ) \end{aligned}$$

with \( \phi \) running over the set of all Gateaux differentiable real functions defined on \( {\mathfrak {g}} \). Here the anti-isotonity is with respect to the pair \( ( \le _{\mathcal C} , \prec ) \) on the set

$$\begin{aligned} \left\{ \phi : {\mathfrak {g}} \rightarrow {\mathbb {R}}: 0 \le _{\mathcal C} \phi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 \right\} . \end{aligned}$$

Proof

By making use of (1), we obtain

$$\begin{aligned} {\mathfrak {t}}_+ ( x + \nabla \varphi (y) )= & {} {\mathfrak {t}}_+ ( x + \nabla \psi (y) + \nabla \varphi (y) - \nabla \psi (y) )\nonumber \\&\prec {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) + {\mathfrak {t}}_+ ( \nabla \varphi (y) - \nabla \psi (y) ). \end{aligned}$$
(9)

The preorder \( \prec \) restricted to \( {\mathfrak {t}}_+ \) is a cone preorder on \( {\mathfrak {t}}_+ \), so we have

$$\begin{aligned} {\mathfrak {t}}_+ ( x + \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ) \prec {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) + {\mathfrak {t}}_+ ( \nabla \varphi (y) - \nabla \psi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ).\nonumber \\ \end{aligned}$$
(10)

For the element \( y \in {\mathfrak {g}} \), there exists a \( g \in G \) such that \( y = g \cdot {\mathfrak {t}}_+ ( y ) \in g \, {\mathfrak {t}}_+ \). Because of the G-increase in the functions \( \psi \) and \( \varphi - \psi \), it follows from (6) that

$$\begin{aligned}&\nabla \psi (y) \in g \, {\mathfrak {t}}_+ ,\\&\nabla \varphi (y) - \nabla \psi (y) = \nabla \left( \varphi - \psi \right) (y) \in g \, {\mathfrak {t}}_+ \end{aligned}$$

[see (6)]. This via (3) ensures that

$$\begin{aligned} {\mathfrak {t}}_+ ( \nabla \psi (y) ) + {\mathfrak {t}}_+ ( \nabla \varphi (y) - \nabla \psi (y) ) = {\mathfrak {t}}_+ ( \nabla \psi (y) + \nabla \varphi (y) - \nabla \psi (y) ) = {\mathfrak {t}}_+ ( \nabla \varphi (y) ). \end{aligned}$$

In consequence, we get

$$\begin{aligned} {\mathfrak {t}}_+ ( \nabla \varphi (y) - \nabla \psi (y) ) = {\mathfrak {t}}_+ ( \nabla \varphi (y) ) - {\mathfrak {t}}_+ ( \nabla \psi (y) ). \end{aligned}$$
(11)

Since

$$\begin{aligned} \nabla \left( \frac{1}{2} \Vert {\cdot }\Vert ^2 \right) (y) = y \end{aligned}$$

and the functions \( \varphi = \psi + ( \varphi - \psi ) \) and \( \nabla \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are G-increasing, we deduce from (6) that

$$\begin{aligned}&\nabla \varphi (y) \in g \, {\mathfrak {t}}_+ ,\\&y - \nabla \varphi (y) = \nabla \left( \frac{1}{2} \Vert {\cdot }\Vert ^2 \right) (y) - \nabla \varphi (y) = \nabla \left( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \right) (y) \in g \, {\mathfrak {t}}_+. \end{aligned}$$

and therefore by (3) we have

$$\begin{aligned} {\mathfrak {t}}_+ ( \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ) = {\mathfrak {t}}_+ ( \nabla \varphi (y) + y - \nabla \psi (y) ) = {\mathfrak {t}}_+ ( y ) , \end{aligned}$$

and next

$$\begin{aligned} {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ) = {\mathfrak {t}}_+ (y) - {\mathfrak {t}}_+ ( \nabla \varphi (y) ). \end{aligned}$$
(12)

Similarly, the function

$$\begin{aligned} \frac{1}{2} \Vert {\cdot }\Vert ^2 - \psi = \left( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \right) + ( \varphi - \psi ) \end{aligned}$$

is G-increasing, because \( \psi \), \( \varphi - \psi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are so. Hence, via (6) we have

$$\begin{aligned}&\nabla \psi (y) \in g \, {\mathfrak {t}}_+ ,\\&y - \nabla \psi (y) = \nabla \left( \frac{1}{2} \Vert {\cdot }\Vert ^2 \right) (y) - \nabla \psi (y) = \nabla \left( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \psi \right) (y) \in g \, {\mathfrak {t}}_+. \end{aligned}$$

This and (3) yield

$$\begin{aligned} {\mathfrak {t}}_+ ( y - \nabla \psi (y) ) = {\mathfrak {t}}_+ ( y ) - {\mathfrak {t}}_+ ( \nabla \psi (y) ). \end{aligned}$$
(13)

Finally, by (11)–(13) and (10) we obtain

$$\begin{aligned}&{\mathfrak {t}}_+ ( x + \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) )\\&\prec {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) + {\mathfrak {t}}_+ ( \nabla \varphi (y) ) - {\mathfrak {t}}_+ ( \nabla \psi (y) ) + {\mathfrak {t}}_+ ( y ) - {\mathfrak {t}}_+ ( \nabla \varphi (y) )\\&\quad = {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) - {\mathfrak {t}}_+ ( \nabla \psi (y) ) + {\mathfrak {t}}_+ ( y ) = {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \psi (y) ) , \end{aligned}$$

as claimed. \(\square \)

As a corollary to Theorem 1, we now present a refinement of the triangle-type inequality (1).

Theorem 2

Let \( \varphi \) be a Gateaux differentiable real function on \( {\mathfrak {g}} \) with continuous gradient \( \nabla \varphi (\cdot )\). Suppose that

$$\begin{aligned} 0 \le _{\mathcal C} \varphi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 , \end{aligned}$$
(14)

i.e., the functions \( \varphi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are G-increasing on \( {\mathfrak {g}} \).

If \( x , y \in {\mathfrak {g}} \) then

$$\begin{aligned} {\mathfrak {t}}_+ ( x + y ) \prec {\mathfrak {t}}_+ ( x + \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ) \prec {\mathfrak {t}}_+ ( x ) + {\mathfrak {t}}_+ ( y ). \end{aligned}$$
(15)

Proof

It follows from (14) that

$$\begin{aligned} 0 \le _{\mathcal C} \psi \le _{\mathcal C} \varphi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 \end{aligned}$$

with \( \psi = 0 \). By applying Theorem 1 with \( \nabla \psi (y) = 0 \), we obtain

$$\begin{aligned}&{\mathfrak {t}}_+ ( x + \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ) \prec {\mathfrak {t}}_+ ( x + \nabla \psi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \psi (y) )\\&\quad = {\mathfrak {t}}_+ ( x ) + {\mathfrak {t}}_+ ( y ). \end{aligned}$$

By virtue of (14), we get

$$\begin{aligned} 0 \le _{\mathcal C} \varphi \le _{\mathcal C} \psi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 \end{aligned}$$

with \( \psi = \frac{1}{2} \Vert {\cdot }\Vert ^2 \). By using Theorem 1 with \( \nabla \psi (y) = y \), we establish

$$\begin{aligned} {\mathfrak {t}}_+ ( x + y )= & {} {\mathfrak {t}}_+ ( x + \nabla \psi (y) )\\&+ {\mathfrak {t}}_+ ( y - \nabla \psi (y) ) \prec {\mathfrak {t}}_+ ( x + \nabla \varphi (y) ) + {\mathfrak {t}}_+ ( y - \nabla \varphi (y) ). \end{aligned}$$

This completes the proof.\(\square \)

Corollary 1

Let \( 0 \le t \le 1 \). If \( x , y \in {\mathfrak {g}} \) then

$$\begin{aligned} {\mathfrak {t}}_+ ( x + y ) \prec {\mathfrak {t}}_+ ( x + t y ) + {\mathfrak {t}}_+ ( y - t y ) \prec {\mathfrak {t}}_+ ( x ) + {\mathfrak {t}}_+ ( y ). \end{aligned}$$
(16)

Proof

The function \( \Vert {\cdot }\Vert = \langle \cdot , \cdot \rangle ^{1/2} \) is G-increasing on \( {\mathfrak {g}} \), because it is convex and G-invariant. Therefore, the functions \( \frac{1}{2} \Vert {\cdot }\Vert ^2 \), \( \varphi = t \frac{1}{2} \Vert {\cdot }\Vert ^2 \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi = ( 1 - t ) \frac{1}{2} \Vert {\cdot }\Vert ^2 \) are G-increasing on \( {\mathfrak {g}} \). That is,

$$\begin{aligned} 0 \le _{\mathcal C} \varphi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2. \end{aligned}$$

So, by Theorem 2 with \( \nabla \varphi (y) = t y \), we infer that inequality (16) holds valid.\(\square \)

3 Results for Real Semisimple Lie Algebras

In this section, we show corresponding results to those in Sect. 2 for a real semisimple Lie algebra \( {\mathfrak {g}} \) with a Cartan decomposition \( {\mathfrak {g}} = {\mathfrak {k}} + {\mathfrak {p}} \), \( {\mathfrak {p}} \ne 0 \), and with a maximal abelian subalgebra \( {{\mathfrak {a}}} \) in \( {\mathfrak {p}} \) equipped with a fixed closed Weyl chamber \( {{\mathfrak {a}}}_+ \) in \( {{\mathfrak {a}}} \). The inner product \( \langle \cdot , \cdot \rangle \) on \( {\mathfrak {p}} \) is the restriction of Killing form, and as previously the norm is given by \( \Vert {\cdot }\Vert = \langle \cdot , \cdot \rangle ^{1/2} \). The preorder \( \prec \) is defined in Sect. 1.

The maximal compact subgroup in the adjoint group \( \mathrm{Int}\,( {\mathfrak {g}} )\) is denoted by K. It acts on \( {\mathfrak {p}} \) by \( \mathrm{Ad}\,K \), i.e., \( k \cdot x = ( \mathrm{Ad}\,k ) x \) for \( k \in K \) and \( x \in {\mathfrak {p}} \).

A function \( \varPhi \) defined on \( {\mathfrak {p}} \) is called K-invariant if

$$\begin{aligned} \varPhi (k \cdot x) = \varPhi (x) \;\; \text{ for } \text{ all } x \in {\mathfrak {p}} \quad \hbox {and}\quad k \in K. \end{aligned}$$

We say that a function \( \varPhi : {\mathfrak {p}} \rightarrow {\mathbb {R}}\) is K-increasing, if for \( x , y \in {\mathfrak {p}} \),

$$\begin{aligned} y \prec x \;\; \text{ implies } \;\; \varPhi (y) \le \varPhi (x). \end{aligned}$$

Here, by definition, the convex cone \( {\mathcal C} \) consists of all K-increasing real functions defined on \( {\mathfrak {p}} \). The cone preorder \( \le _{\mathcal C} \) induced by \( {\mathcal C} \) is defined as follows: for any two real functions \( \varphi : {\mathfrak {p}} \rightarrow {\mathbb {R}}\) and \( \psi : {\mathfrak {p}} \rightarrow {\mathbb {R}}\), we write \( \psi \le _{\mathcal C} \varphi \) provided that the difference function \( \varphi - \psi \) is K-increasing on \( {\mathfrak {p}} \).

Theorem 3

Let \( \varphi \) and \( \psi \) be Gateaux differentiable real functions on \({\mathfrak {p}}\) with continuous gradients \( \nabla \varphi (\cdot )\) and \( \nabla \psi (\cdot )\), respectively. Suppose that

$$\begin{aligned} 0 \le _{\mathcal C} \psi \le _{\mathcal C} \varphi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 , \end{aligned}$$
(17)

i.e., the functions \( \psi \), \( \varphi - \psi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are K-increasing on \( {\mathfrak {p}} \).

If \( x , y \in {\mathfrak {p}} \) then

$$\begin{aligned} {{\mathfrak {a}}}_+ ( x + \nabla \varphi (y) ) + {{\mathfrak {a}}}_+ ( y - \nabla \varphi (y) ) \prec {{\mathfrak {a}}}_+ ( x + \nabla \psi (y) ) + {{\mathfrak {a}}}_+ ( y - \nabla \psi (y) ). \end{aligned}$$
(18)

Proof

The proof of Theorem 3 is similar to that of Theorem 1, and therefore omitted.\(\square \)

An analog of Theorem 2 is the following.

Theorem 4

Let \( \varphi \) be a Gateaux differentiable real function on \( {\mathfrak {p}} \) with continuous gradient \( \nabla \varphi (\cdot )\). Suppose that

$$\begin{aligned} 0 \le _{\mathcal C} \varphi \le _{\mathcal C} \frac{1}{2} \Vert {\cdot }\Vert ^2 , \end{aligned}$$
(19)

i.e., the functions \( \varphi \) and \( \frac{1}{2} \Vert {\cdot }\Vert ^2 - \varphi \) are K-increasing on \( {\mathfrak {p}} \).

If \( x , y \in {\mathfrak {p}} \) then

$$\begin{aligned} {{\mathfrak {a}}}_+ ( x + y ) \prec {{\mathfrak {a}}}_+ ( x + \nabla \varphi (y) ) + {{\mathfrak {a}}}_+ ( y - \nabla \varphi (y) ) \prec {{\mathfrak {a}}}_+ ( x ) + {{\mathfrak {a}}}_+ ( y ). \end{aligned}$$
(20)

Proof

Use a similar method as in the proof of Theorem 2.\(\square \)

Finally, we present an analog of Corollary 1.

Corollary 2

Let \( 0 \le t \le 1 \). If \( x , y \in {\mathfrak {p}} \) then

$$\begin{aligned} {{\mathfrak {a}}}_+ ( x + y ) \prec {{\mathfrak {a}}}_+ ( x + t y ) + {{\mathfrak {a}}}_+ ( y - t y ) \prec {{\mathfrak {a}}}_+ ( x ) + {{\mathfrak {a}}}_+ ( y ). \end{aligned}$$
(21)

Proof

It follows easily from Theorem 4 applied to the function \( \varphi = t \frac{1}{2} \Vert {\cdot }\Vert ^2 \) with \( \nabla \varphi (y) = t y \).\(\square \)