7-13

(i)

ϕ
1

{\displaystyle \phi\_1}

 and

ϕ
2

{\displaystyle \phi\_2}

 are convex in

(
a
,
b
)

{\displaystyle (a,b)}

, for

a
<

x

1

<

x

2

<
b
,
0
≤
θ
≤

{\displaystyle a<x\_{1}<x\_{2}<b,0\leq \theta \leq 1.}

ϕ

1

(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ

ϕ

1

(

x

1

)
(
1
−
θ
)

ϕ

1

x

2

{\displaystyle \phi \_{1}(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta \phi \_{1}(x\_{1})+(1-\theta )\phi \_{1}x\_{2}}

 and

ϕ

2

(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ

ϕ

2

(

x

1

)
(
1
−
θ
)

ϕ

2

x

2

{\displaystyle \phi \_{2}(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta \phi \_{2}(x\_{1})+(1-\theta )\phi \_{2}x\_{2}}

,

(

ϕ

1

ϕ

2

)
(
θ

x

1

(
1
−
θ
)

x

2

# )

ϕ

1

(
θ

x

1

(
1
−
θ
)

x

2

)

ϕ

2

(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ

ϕ

1

(

x

1

)
(
1
−
θ
)

ϕ

1

x

2

θ

ϕ

2

(

x

1

)
(
1
−
θ
)

ϕ

2

x

2

=
θ
(

ϕ

1

ϕ

2

)
(

x

1

)
(
1
−
θ
)
(

ϕ

1

ϕ

2

)

x

2

{\displaystyle (\phi \_{1}+\phi \_{2})(\theta x\_{1}+(1-\theta )x\_{2})=\phi \_{1}(\theta x\_{1}+(1-\theta )x\_{2})+\phi \_{2}(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta \phi \_{1}(x\_{1})+(1-\theta )\phi \_{1}x\_{2}+\theta \phi \_{2}(x\_{1})+(1-\theta )\phi \_{2}x\_{2}=\theta (\phi \_{1}+\phi \_{2})(x\_{1})+(1-\theta )(\phi \_{1}+\phi \_{2})x\_{2}}

Hence,

(

ϕ

1

ϕ

2

)
(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ
(

ϕ

1

ϕ

2

)
(

x

1

)
(
1
−
θ
)
(

ϕ

1

ϕ

2

)

x

2

{\displaystyle (\phi \_{1}+\phi \_{2})(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta (\phi \_{1}+\phi \_{2})(x\_{1})+(1-\theta )(\phi \_{1}+\phi \_{2})x\_{2}}

ϕ

1

ϕ

2

{\displaystyle \phi \_{1}+\phi \_{2}}

 is convex in

(
a
,
b
)

{\displaystyle (a,b)}

.

(ii)

ϕ

{\displaystyle \phi}

 is convex in

(
a
,
b
)

{\displaystyle (a,b)}

,for

a
<

x

1

<

x

2

<
b
,
0
≤
θ
≤

{\displaystyle a<x\_{1}<x\_{2}<b,0\leq \theta \leq 1.}

ϕ
(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ
ϕ
(

x

1

)
(
1
−
θ
)
ϕ

x

2

{\displaystyle \phi (\theta x\_{1}+(1-\theta )x\_{2})\leq \theta \phi (x\_{1})+(1-\theta )\phi x\_{2}}

,

for every positive constant

C

{\displaystyle C}

,

(
c
ϕ
)
(
θ

x

1

(
1
−
θ
)

x

2

# )
c
(
ϕ
)
(
θ

x

1

(
1
−
θ
)

x

2

)
≤
c
(
θ
ϕ
(

x

1

)
(
1
−
θ
)
ϕ

x

2

# )
θ
(
c
ϕ
)
(

x

1

)
(
1
−
θ
)
(
c
ϕ
)

x

2

)

{\displaystyle (c\phi )(\theta x\_{1}+(1-\theta )x\_{2})=c(\phi )(\theta x\_{1}+(1-\theta )x\_{2})\leq c(\theta \phi (x\_{1})+(1-\theta )\phi x\_{2})=\theta (c\phi )(x\_{1})+(1-\theta )(c\phi )x\_{2})}

,

(
c
ϕ
)
(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ
(
c
ϕ
)
(

x

1

)
(
1
−
θ
)
(
c
ϕ
)

x

2

)

{\displaystyle (c\phi )(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta (c\phi )(x\_{1})+(1-\theta )(c\phi )x\_{2})}

c
ϕ

{\displaystyle c\phi }

 is convex in

(
a
,
b
)

{\displaystyle (a,b)}

.

(iii)

ϕ

k

→
ϕ

{\displaystyle \phi \_{k}\rightarrow \phi }

, for all

# k
1
,
2
,
.
.
.

{\displaystyle k=1,2,...}

,

a
<

x

1

<

x

2

<
b
,
0
≤
θ
≤

{\displaystyle a<x\_{1}<x\_{2}<b,0\leq \theta \leq 1.}

ϕ

k

(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ

ϕ

k

(

x

1

)
(
1
−
θ
)

ϕ

k

x

2

{\displaystyle \phi \_{k}(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta \phi \_{k}(x\_{1})+(1-\theta )\phi \_{k}x\_{2}}

,

l
i

m

k
→
∞

ϕ

k

(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ
l
i

m

k
→
∞

ϕ

k

(

x

1

)
(
1
−
θ
)
l
i

m

k
→
∞

ϕ

k

x

2

{\displaystyle lim\_{k\rightarrow \infty }\phi \_{k}(\theta x\_{1}+(1-\theta )x\_{2})\leq \theta lim\_{k\rightarrow \infty }\phi \_{k}(x\_{1})+(1-\theta )lim\_{k\rightarrow \infty }\phi \_{k}x\_{2}}

,

ϕ
(
θ

x

1

(
1
−
θ
)

x

2

)
≤
θ
ϕ
(

x

1

)
(
1
−
θ
)
ϕ

x

2

{\displaystyle \phi (\theta x\_{1}+(1-\theta )x\_{2})\leq \theta \phi (x\_{1})+(1-\theta )\phi x\_{2}}

,

ϕ

{\displaystyle \phi}

 is convex in

(
a
,
b
)

{\displaystyle (a,b)}

.