f ( x ) {\displaystyle f(x)} dom f {\displaystyle \operatorname {dom} f} f ⋆ ( x ⋆ ) {\displaystyle f^{\star }(x^{\star })} dom f ⋆ {\displaystyle \operatorname {dom} f^{\star }} 조건 a f ( x ) {\displaystyle af(x)} dom f {\displaystyle \operatorname {dom} f} a f ⋆ ( x ⋆ / a ) {\displaystyle af^{\star }(x^{\star }/a)} a ⋅ dom f ⋆ {\displaystyle a\cdot \operatorname {dom} f^{\star }} a > 0 {\displaystyle a>0} f ( a x ) {\displaystyle f(ax)} a − 1 ⋅ dom f {\displaystyle a^{-1}\cdot \operatorname {dom} f} f ⋆ ( x ⋆ / a ) {\displaystyle f^{\star }(x^{\star }/a)} a ⋅ dom f ⋆ {\displaystyle a\cdot \operatorname {dom} f^{\star }} a > 0 {\displaystyle a>0} f ( x ) + a {\displaystyle f(x)+a} dom f {\displaystyle \operatorname {dom} f} f ⋆ ( x ⋆ ) − a {\displaystyle f^{\star }(x^{\star })-a} dom f ⋆ {\displaystyle \operatorname {dom} f^{\star }} a ∈ R {\displaystyle a\in \mathbb {R} } f ( x − a ) {\displaystyle f(x-a)} a + dom f {\displaystyle a+\operatorname {dom} f} f ⋆ ( x ⋆ ) + a x ⋆ {\displaystyle f^{\star }(x^{\star })+ax^{\star }} dom f ⋆ {\displaystyle \operatorname {dom} f^{\star }} a ∈ R {\displaystyle a\in \mathbb {R} } f ( x ) + a x {\displaystyle f(x)+ax} dom f {\displaystyle \operatorname {dom} f} f ⋆ ( x ⋆ − a ) {\displaystyle f^{\star }(x^{\star }-a)} a + dom f ⋆ {\displaystyle a+\operatorname {dom} f^{\star }} a ∈ R {\displaystyle a\in \mathbb {R} } f ( x ) + g ( x ) {\displaystyle f(x)+g(x)} dom f ∩ dom g {\displaystyle \operatorname {dom} f\cap \operatorname {dom} g} ( f ⋆ ⋆ inf g ⋆ ) ( x ⋆ ) {\displaystyle (f^{\star }\star _{\text{inf}}g^{\star })(x^{\star })} dom f ⋆ + dom g ⋆ {\displaystyle \operatorname {dom} f^{\star }+\operatorname {dom} g^{\star }} ( f ⋆ inf g ) ( x ) = inf y { f ( x − y ) + g ( y ) } {\displaystyle (f\star _{\text{inf}}g)(x)=\inf _{y}\{f(x-y)+g(y)\}} ( f ⋆ inf g ) ( x ) {\displaystyle (f\star _{\text{inf}}g)(x)} dom f + dom g {\displaystyle \operatorname {dom} f+\operatorname {dom} g} f ⋆ ( x ⋆ ) + g ⋆ ( x ⋆ ) {\displaystyle f^{\star }(x^{\star })+g^{\star }(x^{\star })} dom f ⋆ ∩ dom g ⋆ {\displaystyle \operatorname {dom} f^{\star }\cap \operatorname {dom} g^{\star }} ( f ⋆ inf g ) ( x ) = inf y { f ( x − y ) + g ( y ) } {\displaystyle (f\star _{\text{inf}}g)(x)=\inf _{y}\{f(x-y)+g(y)\}} a x + b {\displaystyle ax+b} R {\displaystyle \mathbb {R} } − b {\displaystyle -b} { a } {\displaystyle \{a\}} | x | p / p {\displaystyle |x|^{p}/p} R {\displaystyle \mathbb {R} } | x ⋆ | p ⋆ / p ⋆ {\displaystyle |x^{\star }|^{p^{\star }}/p^{\star }} R {\displaystyle \mathbb {R} } 1 / p + 1 / p ⋆ = 1 {\displaystyle 1/p+1/p^{\star }=1} , p > 1 {\displaystyle p>1} − x p / p {\displaystyle -x^{p}/p} [ 0 , ∞ ) {\displaystyle [0,\infty )} − | x ⋆ | p ⋆ / p ⋆ {\displaystyle -|x^{\star }|^{p^{\star }}/p^{\star }} ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} 1 / p + 1 / p ⋆ = 1 {\displaystyle 1/p+1/p^{\star }=1} , p < 1 {\displaystyle p<1} exp ( x ) {\displaystyle \exp(x)} R {\displaystyle \mathbb {R} } x ⋆ ( ln ( x ⋆ ) − 1 ) {\displaystyle x^{\star }(\ln(x^{\star })-1)} R + {\displaystyle \mathbb {R} ^{+}} x ln ( x ) {\displaystyle x\ln(x)} R + {\displaystyle \mathbb {R} ^{+}} exp ( x − 1 ) {\displaystyle \exp(x-1)} R {\displaystyle \mathbb {R} } − 1 / 2 − ln x {\displaystyle -1/2-\ln x} R + {\displaystyle \mathbb {R} ^{+}} − 1 / 2 − ln | x ⋆ | {\displaystyle -1/2-\ln |x^{\star }|} R − {\displaystyle \mathbb {R} ^{-}} x exp ( x + 1 ) {\displaystyle x\exp(x+1)} R {\displaystyle \mathbb {R} } x ⋆ ( W ( x ⋆ ) − 1 ) 2 / W ( x ⋆ ) {\displaystyle x^{\star }(W(x^{\star })-1)^{2}/W(x^{\star })} [ − 1 / e , ∞ ) {\displaystyle [-1/e,\infty )} W {\displaystyle W} 는 람베르트 W 함수