Algebraic and Analytic Methods - 1.7 Inequalities

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
1.7.E1 ( ∑ j = 1 n a j ⁢ b j ) 2 ≤ ( ∑ j = 1 n a j 2 ) ⁢ ( ∑ j = 1 n b j 2 ) superscript subscript superscript 𝑛 𝑗 1 subscript 𝑎 𝑗 subscript 𝑏 𝑗 2 subscript superscript 𝑛 𝑗 1 superscript subscript 𝑎 𝑗 2 subscript superscript 𝑛 𝑗 1 superscript subscript 𝑏 𝑗 2 {\displaystyle{\displaystyle\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2}\leq\left% (\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)}}
\left(\sum^{n}_{j=1}a_{j}b_{j}\right)^{2} \leq \left(\sum^{n}_{j=1}a_{j}^{2}\right)\left(\sum^{n}_{j=1}b_{j}^{2}\right)		 

(sum(a[j]*b[j], j = 1..n))^(2) <= (sum((a[j])^(2), j = 1..n))*(sum((b[j])^(2), j = 1..n))
 
((Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None]))^(2) <= (Sum[(Subscript[a, j])^(2), {j, 1, n}, GenerateConditions->None])*(Sum[(Subscript[b, j])^(2), {j, 1, n}, GenerateConditions->None])
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1.7.E2 ∑ j = 1 n a j ⁢ b j ≤ ( ∑ j = 1 n a j p ) 1 / p ⁢ ( ∑ j = 1 n b j q ) 1 / q subscript superscript 𝑛 𝑗 1 subscript 𝑎 𝑗 subscript 𝑏 𝑗 superscript subscript superscript 𝑛 𝑗 1 superscript subscript 𝑎 𝑗 𝑝 1 𝑝 superscript subscript superscript 𝑛 𝑗 1 superscript subscript 𝑏 𝑗 𝑞 1 𝑞 {\displaystyle{\displaystyle\sum^{n}_{j=1}a_{j}b_{j}\leq\left(\sum^{n}_{j=1}a_% {j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}}}
\sum^{n}_{j=1}a_{j}b_{j} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}\left(\sum^{n}_{j=1}b_{j}^{q}\right)^{1/q}		 

sum(a[j]*b[j], j = 1..n) <= (sum((a[j])^(p), j = 1..n))^(1/p)*(sum((b[j])^(q), j = 1..n))^(1/q)
 
Sum[Subscript[a, j]*Subscript[b, j], {j, 1, n}, GenerateConditions->None] <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)*((Sum[(Subscript[b, j])^(q), {j, 1, n}, GenerateConditions->None]))^(1/q)
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1.7.E3 ( ∑ j = 1 n ( a j + b j ) p ) 1 / p ≤ ( ∑ j = 1 n a j p ) 1 / p + ( ∑ j = 1 n b j p ) 1 / p superscript subscript superscript 𝑛 𝑗 1 superscript subscript 𝑎 𝑗 subscript 𝑏 𝑗 𝑝 1 𝑝 superscript subscript superscript 𝑛 𝑗 1 superscript subscript 𝑎 𝑗 𝑝 1 𝑝 superscript subscript superscript 𝑛 𝑗 1 superscript subscript 𝑏 𝑗 𝑝 1 𝑝 {\displaystyle{\displaystyle\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p}% \leq\left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}% \right)^{1/p}}}
\left(\sum^{n}_{j=1}(a_{j}+b_{j})^{p}\right)^{1/p} \leq \left(\sum^{n}_{j=1}a_{j}^{p}\right)^{1/p}+\left(\sum^{n}_{j=1}b_{j}^{p}\right)^{1/p}		 

(sum((a[j]+ b[j])^(p), j = 1..n))^(1/p) <= (sum((a[j])^(p), j = 1..n))^(1/p)+(sum((b[j])^(p), j = 1..n))^(1/p)
 
((Sum[(Subscript[a, j]+ Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p) <= ((Sum[(Subscript[a, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)+((Sum[(Subscript[b, j])^(p), {j, 1, n}, GenerateConditions->None]))^(1/p)
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1.7.E7 H ≤ G 𝐻 𝐺 {\displaystyle{\displaystyle H\leq G}}
H \leq G		 

H <= ((a[1]*a[2] .. a[n])^(1/n))
 
H <= ((Subscript[a, 1]*Subscript[a, 2] \[Ellipsis]Subscript[a, n])^(1/n))
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1.7.E9 M ⁢ ( r ) ≤ M ⁢ ( s ) 𝑀 𝑟 𝑀 𝑠 {\displaystyle{\displaystyle M(r)\leq M(s)}}
M(r) \leq M(s)		 r < s 𝑟 𝑠 {\displaystyle{\displaystyle r<s}} 
((p[1]*(a[1])^(r)+ p[2]*(a[2])^(r)+ .. + p[n]*(a[n])^(r))^(1/r)) <= M(s)
 
((Subscript[p, 1]*(Subscript[a, 1])^(r)+ Subscript[p, 2]*(Subscript[a, 2])^(r)+ \[Ellipsis]+ Subscript[p, n]*(Subscript[a, n])^(r))^(1/r)) <= M[s]
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