LAUSR

In this note, by using gradients of Gateaux differentiable G -increasing functions we prove some refinements of the following Tam’s triangle-type inequality: $$\begin{aligned} {\mathfrak {t}}_+ ( x + y ) \le {\mathfrak {t}}_+ (x) + {\mathfrak {t}}_+ (y) \quad \text{ for }\,\, x ,y \in {\mathfrak {g}} \end{aligned}$$ t + ( x + y ) ≤ t + ( x ) + t + ( y ) for x , y ∈ g in the context of a compact connected Lie group G with Lie algebra $$ {\mathfrak {g}} $$ g and corresponding Weyl chamber $$ {\mathfrak {t}}_+ $$ t + . We also establish refinements of Tam’s inequality: $$\begin{aligned} {{\mathfrak {a}}}_+ ( x + y ) \le {{\mathfrak {a}}}_+ (x) + {{\mathfrak {a}}}_+ (y) \quad \text{ for }\,\, x ,y \in {\mathfrak {p}} \end{aligned}$$ a + ( x + y ) ≤ a + ( x ) + a + ( y ) for x , y ∈ p for a real semisimple Lie algebra $$ {\mathfrak {g}} $$ g with Cartan decomposition $$ {\mathfrak {g}} = {\mathfrak {k}} + {\mathfrak {p}} $$ g = k + p , a maximal abelian subalgebra $$ {{\mathfrak {a}}} $$ a in $$ {\mathfrak {p}} $$ p and its closed Weyl chamber $$ {{\mathfrak {a}}}_+ $$ a + .