Question Number 1928 by Yozzi last updated on 24/Oct/15 | ||
$${Prove}\:{that},\:{if}\:{p}>{q}>\mathrm{0}\:{and}\:{x}\geqslant\mathrm{0},\:{then} \\ $$$$\:\:\:\:\:\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right).\: \\ $$ | ||
Commented by Rasheed Soomro last updated on 24/Oct/15 | ||
$${Prove}\:{that},\:{if}\:{p}>{q}>\mathrm{0}\:{and}\:{x}\geqslant\mathrm{0},\:{then} \\ $$$$\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right).\: \\ $$$$−−−−−−−−−−−−−−−− \\ $$$${A}\:{Try}.\:{Not}\:{much}\:{confident} \\ $$$${Let}'{s}\:{try}\:{to}\:{achieve}\:{simpler}\:{equivalent}\:{of} \\ $$$${the}\:{given}\:{statment},{which}\:{is}\:{easy}\:{to}\:{prove}. \\ $$$${Let}\:\:{p}={q}+{k},\:{where}\:{k}>\mathrm{0} \\ $$$$\frac{\mathrm{1}}{{q}+{k}}\left(\frac{{x}^{{q}+{k}} }{{q}+{k}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$$\frac{\mathrm{1}}{{q}+{k}}\left(\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}+{k}+\mathrm{1}}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} −{q}−\mathrm{1}}{{q}+\mathrm{1}}\right) \\ $$$$\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{\left({q}+{k}\right)\left({q}+{k}+\mathrm{1}\right)}\geqslant\frac{{x}^{{q}} −{q}−\mathrm{1}}{{q}\left({q}+\mathrm{1}\right)} \\ $$$$\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}+\mathrm{2}{qk}+{k}^{\mathrm{2}} +{k}}\geqslant\frac{{x}^{{q}} −{q}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}} \\ $$$${q}>\mathrm{0}\:\wedge\:{k}>\mathrm{0}\:\Rightarrow\:\mathrm{2}{qk}+{k}^{\mathrm{2}} +{k}>\mathrm{0} \\ $$$${Reduction}\:{of}\:\:\mathrm{2}{qk}+{k}^{\mathrm{2}} +{k}\:\left({a}\:+{ve}\:{number}\right) \\ $$$${in}\:{the}\:{denominator}\:{of}\:{greater}/{equal}\:\:{side} \\ $$$${will}\:{leave}\:{no}\:{effect}\:{on}\:{the}\:{inequality}. \\ $$$${Hence}, \\ $$$$\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}}\geqslant\frac{{x}^{{q}} −{q}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}} \\ $$$${Multiplying}\:{by}\:{q}^{\mathrm{2}} +{q}\:\left(>\mathrm{0}\right)\:{to}\:{both}\:{sides} \\ $$$${x}^{{q}+{k}} −{q}−{k}−\mathrm{1}\geqslant{x}^{{q}} −{q}−\mathrm{1} \\ $$$${x}^{{q}+{k}} −{k}\geqslant{x}^{{q}} \\ $$$${Since}\:{k}>\mathrm{0} \\ $$$${x}^{{q}+{k}} −{k}+{k}\geqslant{x}^{{q}} \\ $$$${x}^{{q}} .{x}^{{k}} \geqslant{x}^{{q}} \\ $$$${Given}\:{that}\:{x}>\mathrm{0}\:{or}\:{x}=\mathrm{0} \\ $$$${for}\:{x}>\mathrm{0},\:{x}^{{q}} \:{can}\:{be}\:{cancelled}\:{from}\:{both}\:{sides}: \\ $$$${x}^{{k}} \geqslant\mathrm{1} \\ $$$${Let}'{s}\:{prove}\:{for}\:{x}\geqslant\mathrm{1}\:{first}. \\ $$$$\:{x}^{{k}} \geqslant\mathrm{1}\:\:{is}\:{completely}\:{equivalent}\:{of} \\ $$$${the}\:{given}\:{statement}\:{and}\:{it}\:{is}\:{clearly}\:{true}\:{since} \\ $$$${x}\geqslant\mathrm{1}\:\wedge\:{k}>\mathrm{0} \\ $$$${Hence}\:{given}\:{statement}\:{is}\:{true}\:{for}\:{x}\geqslant\mathrm{1} \\ $$$${For}\:{x}=\mathrm{0}\:{the}\:{result}\:{can}\:{be}\:{proved}\:{directly}. \\ $$$${Now}\:{the}\:{problem}\:{is}\:{for}\:{x}\in\left(\mathrm{0},\mathrm{1}\right) \\ $$$$.... \\ $$$${Continue} \\ $$ | ||
Commented by prakash jain last updated on 24/Oct/15 | ||
$$\mathrm{Observation}\:\mathrm{on}\:\mathrm{Rasheed}\:\mathrm{Comment} \\ $$$$\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}+\mathrm{2}{qk}+{k}^{\mathrm{2}} +{k}}\geqslant\frac{{x}^{\mathrm{2}} −{q}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}} \\ $$$$\mathrm{is}\:\mathrm{not}\:\mathrm{equivalent}\:\mathrm{to}\:\left(\mathrm{if}\:\mathrm{2}{qk}+{k}^{\mathrm{2}} +{k}>\mathrm{0}\right) \\ $$$$\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}}\geqslant\frac{{x}^{\mathrm{2}} −{q}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}} \\ $$$$\:\mathrm{since}\:\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}+\mathrm{2}{qk}+{k}^{\mathrm{2}} +{k}}\:<\:\frac{{x}^{{q}+{k}} −{q}−{k}−\mathrm{1}}{{q}^{\mathrm{2}} +{q}} \\ $$$$\mathrm{Suggestion}:\:\mathrm{inequality}\:\mathrm{may}\:\mathrm{be}\:\mathrm{simpler}\:\mathrm{to}\:\mathrm{prove} \\ $$$$\mathrm{if}\:\mathrm{you}\:\mathrm{try}\:{p}={kq},\:{k}>\mathrm{1} \\ $$ | ||
Commented by Rasheed Soomro last updated on 24/Oct/15 | ||
$$\boldsymbol{\mathcal{THANKS}\mathrm{S}}\mathrm{S}{SSsss}...\:{for}\:{correction}\:{and}\:{suggestion}! \\ $$ | ||
Commented by Rasheed Soomro last updated on 25/Oct/15 | ||
$${Prove}\:{that},\:{if}\:{p}>{q}>\mathrm{0}\:{and}\:{x}\geqslant\mathrm{0},\:{then} \\ $$$$\:\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right).\: \\ $$$$−−−−−×××−−−−−−− \\ $$$${To}\:{get}\:{a}\:\boldsymbol{\mathrm{simpler}}\:\boldsymbol{\mathrm{equivalent}}\left(?\right): \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}}......................\left(\mathrm{1}\right) \\ $$$${Now},\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{p}>{q}\:\Rightarrow\:\frac{\mathrm{1}}{{p}}<\frac{\mathrm{1}}{{q}}\:\Rightarrow−\frac{\mathrm{1}}{{p}}>−\frac{\mathrm{1}}{{q}}..........\left(\mathrm{2}\right) \\ $$$${Adding}\:\left(\mathrm{1}\right)\:{and}\:\left(\mathrm{2}\right) \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}\:\geqslant\:\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)} \\ $$$${Is}\:{this}\:{equivalent}\:{to}\:{the}\:{original}? \\ $$$${Does}\:{adding}\:{a}\:\boldsymbol{\mathrm{same}}−\boldsymbol{\mathrm{sense}}\:\boldsymbol{\mathrm{inequality}}\:{yield}\:{an} \\ $$$$\boldsymbol{\mathrm{equivalent}}\:\boldsymbol{\mathrm{inequality}}? \\ $$$${Let}\:{p}={qk},\:{where}\:{k}>\mathrm{1} \\ $$$$\frac{{x}^{{qk}} }{{qk}\left({qk}+\mathrm{1}\right)}\:\geqslant\:\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)} \\ $$$$\frac{{x}^{{qk}} }{{k}\left({qk}+\mathrm{1}\right)}\:\geqslant\:\frac{{x}^{{q}} }{{q}+\mathrm{1}}\:\:\:\:\:\:\:\:\:\:\left[{Multiplying}\:{by}\:{q}\left(>\mathrm{0}\right)\:\right] \\ $$$$\frac{{x}^{{qk}} }{{qk}^{\mathrm{2}} +{k}}\:\geqslant\:\frac{{x}^{{q}} }{{q}+\mathrm{1}}\: \\ $$$$ \\ $$$$\boldsymbol{\mathrm{Continue}} \\ $$ | ||
Commented by Rasheed Soomro last updated on 25/Oct/15 | ||
$$\boldsymbol{\mathrm{On}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{comment}}\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{prakash}}\:\boldsymbol{\mathrm{jain}} \\ $$$$\boldsymbol{\mathrm{Sir}}\:{you}\:{have}\:{said}\:{that}\:{a}>{b}\:{is}\:{not}\:{equivalent}\:{to}\:{A}>{B} \\ $$$${because}\:\left({after}\:{dropping}\:{some}\:+{ve}\:{value}\:{from}\:{denominator}\right) \\ $$$${A}<{a}. \\ $$$${Does}\:{this}\:{mean}\:{a}>{b}\:{and}\:{A}>{B}\:{will}\:{be}\:{equivalent}\:{only} \\ $$$${when}\:{A}={a}? \\ $$ | ||
Commented by prakash jain last updated on 25/Oct/15 | ||
$$\mathrm{While}\:\mathrm{proving} \\ $$$$\mathrm{if}\:{a}<{A}\:\mathrm{then}\:{a}>{b}\Rightarrow\:{A}>\mathrm{b}\:\mathrm{but}\:{A}>{b}\nRightarrow{a}>{b} \\ $$$$\mathrm{if}\:{a}>{A}\:\mathrm{then}\:{A}>\mathrm{b}\Rightarrow{a}>{b}\:\mathrm{but}\:{a}>{b}\nRightarrow{A}>{b} \\ $$ | ||
Commented by prakash jain last updated on 25/Oct/15 | ||
$$\mathrm{Rasheed}\:\mathrm{regarding}\:\mathrm{equality}\:\mathrm{for}\:{p}>{q} \\ $$$$\mathrm{If}\:\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}\:\:......\left(\mathrm{1}\right) \\ $$$$\mathrm{then}\:\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}\:−\frac{\mathrm{1}}{{q}}\:......\left(\mathrm{2}\right) \\ $$$$\left(\mathrm{1}\right)\:\Rightarrow\left(\mathrm{2}\right)\:\mathrm{however}\:\left(\mathrm{2}\right)\:\mathrm{does}\:\mathrm{not}\:\mathrm{imply}\:\left(\mathrm{1}\right). \\ $$$$\mathrm{Now}\:\mathrm{you}\:\mathrm{cannot}\:\mathrm{prove}\:\mathrm{1}\:\mathrm{since}\:\mathrm{is}\:\mathrm{not}\:\mathrm{true}. \\ $$$$\mathrm{Try}\:{x}=\mathrm{1},{p}=\mathrm{2},\:{q}=\mathrm{1} \\ $$ | ||
Commented by Rasheed Soomro last updated on 25/Oct/15 | ||
$$\boldsymbol{\mathcal{THANKS}}\:\boldsymbol{{for}}\:\boldsymbol{{so}}\:\boldsymbol{{many}}\:\boldsymbol{{valueable}}\:\boldsymbol{{explanations}}! \\ $$$$\mathrm{However}\:\mathrm{I}\:\mathrm{would}\:\mathrm{like}\:\mathrm{to}\:\mathrm{understand}\:\mathrm{clearly} \\ $$$$\mathrm{why} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}} \\ $$$$\mathrm{does}\:\mathrm{not}\:\underset{−} {\mathrm{imply}}\:\mathrm{to} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\frac{{x}^{{p}} }{\mathrm{p}\left({p}+\mathrm{1}\right)}\geqslant\frac{{x}^{{q}} }{\mathrm{q}\left({q}+\mathrm{1}\right)}\:\:\:\: \\ $$$$\mathrm{when}\:\mathrm{p}>\mathrm{q}>\mathrm{0}\:\mathrm{is}\:\mathrm{also}\:\mathrm{given}\:? \\ $$$$\:\:\:\:\:\:\:\:\mathrm{p}>\mathrm{q}\:\Rightarrow\frac{\mathrm{1}}{{p}}<\frac{\mathrm{1}}{{q}}\Rightarrow−\frac{\mathrm{1}}{{p}}>−\frac{\mathrm{1}}{{q}} \\ $$$$\:\:\frac{{x}^{{p}} }{\mathrm{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{\mathrm{q}\left({q}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}}\:\wedge\:−\frac{\mathrm{1}}{{p}}>−\frac{\mathrm{1}}{{q}} \\ $$$$\Rightarrow\frac{{x}^{{p}} }{\mathrm{p}\left({p}+\mathrm{1}\right)}\geqslant\frac{{x}^{{q}} }{\mathrm{q}\left({q}+\mathrm{1}\right)}\:\:\left[{Subtracting}\:{second}\:{inequality}\:{from}\:{first}\right] \\ $$$${However}\:{as}\:{for}\:{as}\:\underset{−} {{equivalence}}\:{is}\:{concerned}\:{I}\:{admit} \\ $$$${that}\:\boldsymbol{{these}}\:\boldsymbol{{two}}\:\boldsymbol{{are}}\:\boldsymbol{{not}}\:\boldsymbol{{equivalent}}.{Perhaps}\:{this}\:{is} \\ $$$${the}\:{reason}\:{that}\:{one}\:{satisfy}\:{some}\:{values}\:{and}\:{other}\:{not}. \\ $$$${But}\:{since}\:{the}\:{goal}\:{was}\:{to}\:{determine}\:{an}\:{equivalent}\:{so} \\ $$$$\boldsymbol{{I}}\:\boldsymbol{{failed}}\:\boldsymbol{{to}}\:\boldsymbol{{achieve}}. \\ $$$$\boldsymbol{{I}}\:\boldsymbol{{think}}\:\boldsymbol{{implication}}\:\boldsymbol{{and}}\:\boldsymbol{{equivalence}}\:\boldsymbol{{are}}\:\boldsymbol{{two}} \\ $$$$\boldsymbol{{different}}\:\boldsymbol{{things}}. \\ $$$$\boldsymbol{{Anyway}}\:\boldsymbol{{your}}\:\boldsymbol{{opinion}}\:\boldsymbol{{will}}\:\boldsymbol{{be}}\:\boldsymbol{{an}}\:\boldsymbol{{expert}}'\boldsymbol{{s}}\:\boldsymbol{{opinion}}\: \\ $$$$\boldsymbol{{and}}\:\boldsymbol{{my}}\:\boldsymbol{{question}}\:\boldsymbol{{is}}\:\boldsymbol{{a}}\:\boldsymbol{{student}}'\boldsymbol{{s}}\:\boldsymbol{{queztion}}. \\ $$$$\boldsymbol{{I}}\:\boldsymbol{{am}}\:\boldsymbol{{here}}\:\boldsymbol{{to}}\:\boldsymbol{{learn}}\:\boldsymbol{{from}}\:\boldsymbol{{you}}\:\boldsymbol{{and}}\:\boldsymbol{{other}}\:\boldsymbol{{experts}}. \\ $$$$ \\ $$ | ||
Commented by Rasheed Soomro last updated on 27/Oct/15 | ||
$${Prove}\:{that},\:{if}\:{p}>{q}>\mathrm{0}\:{and}\:{x}\geqslant\mathrm{0},\:{then} \\ $$$$\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$$−×−×−×−×−×−×−×−×−×− \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}} \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}\geqslant\frac{\mathrm{1}}{{p}}−\frac{\mathrm{1}}{{q}} \\ $$$$\frac{{x}^{{p}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} {p}\left({p}+\mathrm{1}\right)}{{pq}\left({p}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant\frac{{q}−{p}}{{pq}}.....................\left({A}\right) \\ $$$${Approach}\left(\mathrm{1}\right) \\ $$$$\underset{−} {{Let}\:{p}={qk},{where}\:{k}>\mathrm{1}} \\ $$$$\frac{{x}^{{p}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} {p}\left({p}+\mathrm{1}\right)}{{pq}\left({p}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant\frac{{q}−{p}}{{pq}}.....................\left({A}\right) \\ $$$$\frac{{x}^{{qk}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} {qk}\left({qk}+\mathrm{1}\right)}{{q}^{\mathrm{2}} {k}\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant\frac{{q}−{qk}}{{q}^{\mathrm{2}} {k}} \\ $$$$\frac{{x}^{{qk}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} {qk}\left({qk}+\mathrm{1}\right)}{\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant{q}\left(\mathrm{1}−{k}\right) \\ $$$$\frac{{x}^{{qk}} \left({q}+\mathrm{1}\right)−{x}^{{q}} {k}\left({qk}+\mathrm{1}\right)}{\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant\mathrm{1}−{k} \\ $$$$\frac{{x}^{{qk}} \left({q}+\mathrm{1}\right)−{x}^{{q}} {k}\left({qk}+\mathrm{1}\right)}{\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}+{k}\geqslant\mathrm{1} \\ $$$$\frac{{x}^{{qk}} \left({q}+\mathrm{1}\right)−{x}^{{q}} {k}\left({qk}+\mathrm{1}\right)+{k}\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}{\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\:\geqslant\:\mathrm{1} \\ $$$$\frac{{x}^{{qk}} {q}+{x}^{{qk}} −{x}^{{q}} {qk}^{\mathrm{2}} −{x}^{{q}} {k}+{q}^{\mathrm{2}} {k}^{\mathrm{2}} +{qk}^{\mathrm{2}} +{qk}+{k}}{\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant\mathrm{1} \\ $$$$\frac{{x}^{{qk}} {q}+{x}^{{qk}} −{x}^{{q}} {qk}^{\mathrm{2}} −{x}^{{q}} {k}+{q}^{\mathrm{2}} {k}^{\mathrm{2}} +{qk}^{\mathrm{2}} +{qk}+{k}}{{q}^{\mathrm{2}} {k}+{qk}+{q}+\mathrm{1}}\geqslant\mathrm{1} \\ $$$${x}^{{qk}} {q}+{x}^{{qk}} −{x}^{{q}} {qk}^{\mathrm{2}} −{x}^{{q}} {k}+{q}^{\mathrm{2}} {k}^{\mathrm{2}} +{qk}^{\mathrm{2}} +{qk}+{k}−{q}^{\mathrm{2}} {k}−{qk}−{q}−\mathrm{1}\geqslant\mathrm{0} \\ $$$$...... \\ $$$${Approach}\left(\mathrm{2}\right) \\ $$$${Let}\:{p}={q}+{k},{where}\:{k}>\mathrm{0} \\ $$$$\frac{{x}^{{p}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} {p}\left({p}+\mathrm{1}\right)}{{pq}\left({p}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)}\geqslant\frac{{q}−{p}}{{pq}}.....................\left({A}\right) \\ $$$$\frac{{x}^{\left({q}+{k}\right)} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} \left({q}+{k}\right)\left\{\left({q}+{k}\right)+\mathrm{1}\right\}}{{q}\left({q}+\mathrm{1}\right)\left\{\left({q}+{k}\right)+\mathrm{1}\right\}\left({q}+\mathrm{1}\right)}\geqslant\frac{{q}−\left({q}+{k}\right)}{{q}\left({q}+{k}\right)} \\ $$$$\frac{{x}^{{q}+{k}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} \left({q}+{k}\right)\left({q}+{k}+\mathrm{1}\right)}{{q}\left({q}+\mathrm{1}\right)^{\mathrm{2}} \left({q}+{k}+\mathrm{1}\right)}\geqslant\frac{−{k}}{{q}\left({q}+{k}\right)} \\ $$$$\frac{{x}^{{q}+{k}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} \left({q}+{k}\right)\left({q}+{k}+\mathrm{1}\right)}{\left({q}+\mathrm{1}\right)^{\mathrm{2}} \left({q}+{k}+\mathrm{1}\right)}\geqslant\frac{−{k}}{{q}+{k}}\:\:\:\:\left[{Multiply}\:{by}\:{q}\left(>\mathrm{0}\right)\right] \\ $$$$\frac{{x}^{{q}+{k}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} \left({q}+{k}\right)\left({q}+{k}+\mathrm{1}\right)}{\left({q}+\mathrm{1}\right)^{\mathrm{2}} \left({q}+{k}+\mathrm{1}\right)}+\frac{{k}}{{q}+{k}}\:\geqslant\mathrm{0}\:\:\: \\ $$$$\frac{\left\{{x}^{{q}+{k}} {q}\left({q}+\mathrm{1}\right)−{x}^{{q}} \left({q}+{k}\right)\left({q}+{k}+\mathrm{1}\right)\right\}\left({q}+{k}\right)+{k}\left({q}+\mathrm{1}\right)^{\mathrm{2}} \left({q}+{k}+\mathrm{1}\right)}{\left({q}+\mathrm{1}\right)^{\mathrm{2}} \left({q}+{k}\right)\left({q}+{k}+\mathrm{1}\right)}\:\geqslant\mathrm{0}\:\:\: \\ $$$${Continue} \\ $$ | ||
Commented by Rasheed Soomro last updated on 26/Oct/15 | ||
$$\:\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$${p}>\mathrm{0},{q}>\mathrm{0}\Rightarrow{pq}>\mathrm{0},{multiplying}\:{by}\:{pq}\:{to}\:{both}\:{sides} \\ $$$$\:{q}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant{p}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$$\:\frac{{qx}^{{p}} }{{p}+\mathrm{1}}−{q}\geqslant\frac{{px}^{{q}} }{{q}+\mathrm{1}}−{p}_{} \\ $$$$\:\frac{{qx}^{{p}} }{{p}+\mathrm{1}}+{p}\geqslant\frac{{px}^{{q}} }{{q}+\mathrm{1}}+{q}\:\:\left[{Adding}\:{p}+{q}\:{to}\:{both}\:{sides}\right] \\ $$$$\:\frac{{qx}^{{p}} +{p}\left({p}+\mathrm{1}\right)}{{p}+\mathrm{1}}\geqslant\frac{{px}^{{q}} +{q}\left({q}+\mathrm{1}\right)}{{q}+\mathrm{1}} \\ $$$${Let}\:{p}={qk},\:{where}\:{k}>\mathrm{1} \\ $$$$\:\frac{{qx}^{{qk}} +{qk}\left({qk}+\mathrm{1}\right)}{{qk}+\mathrm{1}}\geqslant\frac{{qkx}^{{q}} +{q}\left({q}+\mathrm{1}\right)}{{q}+\mathrm{1}} \\ $$$${q}>\mathrm{0}\:\wedge\:{k}>\mathrm{0}\Rightarrow{q}+\mathrm{1}>\mathrm{0}\:\wedge\:{qk}+\mathrm{1}>\mathrm{0}\:\Rightarrow\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)>\mathrm{0} \\ $$$${Multiplying}\:{by}\:\left({qk}+\mathrm{1}\right)\left({q}+\mathrm{1}\right)\:{to}\:{both}\:{sides}: \\ $$$$\left.{q}\left\{{x}^{{qk}} +{qk}^{\mathrm{2}} +{k}\right\}\left({q}+\mathrm{1}\right)\geqslant{q}\left\{{kx}^{{q}} +{q}+\mathrm{1}\right\}\left({qk}+\mathrm{1}\right)\:\:\right\}\boldsymbol{\div}{q}\:\left(>\mathrm{0}\right) \\ $$$$\left\{{x}^{{qk}} +{qk}^{\mathrm{2}} +{k}\right\}\left({q}+\mathrm{1}\right)\geqslant\left\{{kx}^{{q}} +{q}+\mathrm{1}\right\}\left({qk}+\mathrm{1}\right) \\ $$$${qx}^{{qk}} +{q}^{\mathrm{2}} {k}^{\mathrm{2}} +\overset{×} {{qk}}+{x}^{{qk}} +{qk}^{\mathrm{2}} +{k}\geqslant{qk}^{\mathrm{2}} {x}^{{q}} +{q}^{\mathrm{2}} {k}+\overset{×} {{qk}}+{kx}^{{q}} +{q}+\mathrm{1} \\ $$$${qx}^{{qk}} +{q}^{\mathrm{2}} {k}^{\mathrm{2}} +{x}^{{qk}} +{qk}^{\mathrm{2}} +{k}\geqslant{qk}^{\mathrm{2}} {x}^{{q}} +{q}^{\mathrm{2}} {k}+{kx}^{{q}} +{q}+\mathrm{1} \\ $$$$\left({q}+\mathrm{1}\right){x}^{{qk}} +{q}^{\mathrm{2}} {k}^{\mathrm{2}} +{qk}^{\mathrm{2}} +{k}\geqslant{k}\left({qk}+\mathrm{1}\right){x}^{{q}} +{q}^{\mathrm{2}} {k}+{q}+\mathrm{1} \\ $$$${The}\:{above}\:{inequality}\:{is}\:{equivalent}\:{to}\:{the}\:{inequality} \\ $$$${to}\:{be}\:{proved}.{To}\:{prove}\:{the}\:{given}\:{it}\:{is}\:{suficient}\:{to} \\ $$$${prove}\:{its}\:{equivalent}. \\ $$$${If}\:{we}\:{could}\:{prove}\:{the}\:{following}\:{inequalities} \\ $$$${They}\:{imply}\:{aove}\:{inequality}.\:{Howdver}\:\:{vice}\:{versa} \\ $$$${is}\:{not}\:{correct}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({q}+\mathrm{1}\right){x}^{{qk}} \geqslant{k}\left({qk}+\mathrm{1}\right){x}^{{q}} \:\:\left[.....\right].....................\left({i}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{q}^{\mathrm{2}} {k}^{\mathrm{2}} \geqslant{q}^{\mathrm{2}} {k}\Rightarrow\left({q}^{\mathrm{2}} {k}\right){k}\geqslant{q}^{\mathrm{2}} {k}\:\left[\because\:{k}>\mathrm{1}\right]..........\left({ii}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{qk}^{\mathrm{2}} \geqslant{q}\:\left[{k}>\mathrm{1}\:\wedge\:{q}>\mathrm{0}\Rightarrow{k}^{\mathrm{2}} >\mathrm{1}\Rightarrow{qk}^{\mathrm{2}} \geqslant{q}.....\left({iii}\right)\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{k}\geqslant\mathrm{1}\:\left[{Assumption}\right]...........................\left({iv}\right) \\ $$$${Unfortinuately}\:\left({i}\right)\:\:{is}\:{not}\:{always}\:{correct}.{For}\:{example} \\ $$$${at}\:{x}=\mathrm{1}\:{it}\:{is}\:{false}! \\ $$$${Dilli}\:{hunooz}\:{door}\:{ast} \\ $$$${Continue} \\ $$ | ||
Answered by Rasheed Soomro last updated on 26/Oct/15 | ||
$${Experiment} \\ $$$$\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$${Let}\:{p}={q}^{{k}} ,{where}\:{k}>\mathrm{1} \\ $$$$\frac{\mathrm{1}}{{q}^{{k}} }\left(\frac{{x}^{{q}^{{k}} } }{{q}^{{k}} +\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$$\frac{\mathrm{1}}{{q}^{{k}−\mathrm{1}} }\left(\frac{{x}^{{q}^{{k}} } }{{q}^{{k}} +\mathrm{1}}−\mathrm{1}\right)\geqslant\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$$\frac{{x}^{{q}^{{k}} } }{{q}^{{q}−\mathrm{1}} \left({q}^{{k}} +\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}^{{k}−\mathrm{1}} }\geqslant\frac{{x}^{{q}} −{q}−\mathrm{1}}{{q}+\mathrm{1}} \\ $$$$\frac{{x}^{{q}^{{k}} } −{q}^{{k}} −\mathrm{1}}{{q}^{{k}−\mathrm{1}} \left({q}^{{k}} +\mathrm{1}\right)}\geqslant\frac{{x}^{{q}} −{q}−\mathrm{1}}{{q}+\mathrm{1}} \\ $$$$\left({x}^{{q}^{{k}} } −{q}^{{k}} −\mathrm{1}\right)\left({q}+\mathrm{1}\right)\geqslant{q}^{{k}−\mathrm{1}} \left({q}^{{k}} +\mathrm{1}\right)\left({x}^{{q}} −{q}−\mathrm{1}\right) \\ $$$${qx}^{{q}^{{k}} } −{q}^{{k}+\mathrm{1}} −{q}+{x}^{{q}^{{k}} } −{q}^{{k}} −\mathrm{1}\geqslant{q}^{{k}−\mathrm{1}} \left({q}^{{k}} {x}^{{q}} −{q}^{{k}+\mathrm{1}} −{q}^{{k}} +{x}^{{q}} −{q}−\mathrm{1}\right) \\ $$$$\left.{qx}^{{q}^{{k}} } −{q}^{{k}+\mathrm{1}} −{q}+{x}^{{q}^{{k}} } −{q}^{{k}} −\mathrm{1}\geqslant{q}^{\mathrm{2}{k}−\mathrm{1}} {x}^{{q}} −{q}^{\mathrm{2}{k}} −{q}^{\mathrm{2}{k}−\mathrm{1}} +{q}^{{k}−\mathrm{1}} {x}^{{q}} −{q}^{{k}} −{q}^{{k}−\mathrm{1}} \right) \\ $$$$\left.\left({q}+\mathrm{1}\right){x}^{{q}^{{k}} } −{q}^{{k}+\mathrm{1}} −{q}−{q}^{{k}} −\mathrm{1}\geqslant\left({q}^{\mathrm{2}{k}−\mathrm{1}} +{q}^{{k}−\mathrm{1}} \right){x}^{{q}} −{q}^{\mathrm{2}{k}} −{q}^{\mathrm{2}{k}−\mathrm{1}} −{q}^{{k}} −{q}^{{k}−\mathrm{1}} \right) \\ $$ | ||
Answered by Rasheed Soomro last updated on 28/Oct/15 | ||
$$\frac{\mathrm{1}}{{p}}\left(\frac{{x}^{{p}} }{{p}+\mathrm{1}}−\mathrm{1}\right)\geqslant\frac{\mathrm{1}}{{q}}\left(\frac{{x}^{{q}} }{{q}+\mathrm{1}}−\mathrm{1}\right) \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{p}}\geqslant\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}−\frac{\mathrm{1}}{{q}} \\ $$$$\frac{{x}^{{p}} }{{p}\left({p}+\mathrm{1}\right)}−\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}\geqslant\frac{\mathrm{1}}{{p}}−\frac{\mathrm{1}}{{q}} \\ $$$${Let}\:\:{p}={qk},{where}\:{k}>\mathrm{1} \\ $$$$\frac{{x}^{{qk}} }{{qk}\left({qk}+\mathrm{1}\right)}−\frac{{x}^{{q}} }{{q}\left({q}+\mathrm{1}\right)}\geqslant\frac{\mathrm{1}}{{qk}}−\frac{\mathrm{1}}{{q}} \\ $$$$\frac{{x}^{{qk}} }{{k}\left({qk}+\mathrm{1}\right)}−\frac{{x}^{{q}} }{\left({q}+\mathrm{1}\right)}\geqslant\frac{\mathrm{1}}{{k}}−\mathrm{1}\:\:\:\left[{Multkplying}\:{by}\:{q}\left(>\mathrm{0}\right)\right] \\ $$$$\frac{{x}^{{qk}} \left({q}+\mathrm{1}\right)−{kx}^{{q}} \left({qk}+\mathrm{1}\right)}{{k}\left({q}+\mathrm{1}\right)\left({qk}+\mathrm{1}\right)}\geqslant\frac{\mathrm{1}−{k}}{{k}} \\ $$$$\frac{{x}^{{qk}} \left({q}+\mathrm{1}\right)−{kx}^{{q}} \left({qk}+\mathrm{1}\right)}{\left({q}+\mathrm{1}\right)\left({qk}+\mathrm{1}\right)}\geqslant\mathrm{1}−{k}\:\:\:\:\left[{Multiplying}\:{by}\:{q}\left(>\mathrm{0}\right)\right] \\ $$$${x}^{{qk}} \left({q}+\mathrm{1}\right)−{kx}^{{q}} \left({qk}+\mathrm{1}\right)\geqslant\left(\mathrm{1}−{k}\right)\left({q}+\mathrm{1}\right)\left({qk}+\mathrm{1}\right) \\ $$$${qx}^{{ak}} +{x}^{{qk}} −{qk}^{\mathrm{2}} {x}^{{q}} −{kx}^{{q}} \geqslant\left(\mathrm{1}−{k}\right)\left({q}^{\mathrm{2}} {k}+{q}+{qk}+\mathrm{1}\right) \\ $$$${qx}^{{ak}} +{x}^{{qk}} −{qk}^{\mathrm{2}} {x}^{{q}} −{kx}^{{q}} \geqslant{q}^{\mathrm{2}} {k}+{q}+\overset{×} {{qk}}+\mathrm{1}−{q}^{\mathrm{2}} {k}^{\mathrm{2}} −\overset{×} {{qk}}−{qk}^{\mathrm{2}} −{k} \\ $$$${qx}^{{ak}} +{x}^{{qk}} −{qk}^{\mathrm{2}} {x}^{{q}} −{kx}^{{q}} \geqslant{q}^{\mathrm{2}} {k}+{q}+\mathrm{1}−{q}^{\mathrm{2}} {k}^{\mathrm{2}} −{qk}^{\mathrm{2}} −{k} \\ $$$$ \\ $$$$ \\ $$ | ||