Criterion for all derivatives of a function dominating the derivatives of another function
up vote
0
down vote
favorite
Given two positive functions $fcolon mathbb{R} rightarrow mathbb{R}^{+}$ and $gcolon mathbb{R} rightarrow mathbb{R}^{+}$ and a point $ x in mathbb{R}$.
Is there some sufficient (and preferably necessary) criterion asserting that every $n$-th derivative of $g$ dominates the respective $n$-th derivative of $f$ at point $x$?
I.e. the criterion should assert that
$$
forall ncolon quad f^{(n)}(x) leq g^{(n)}(x)~.
$$
(Needless to say, the criterion should be somehow easier to check than the above.)
real-analysis derivatives
asked Nov 21 at 23:36
blk
356
add a comment |
up vote
0
down vote
favorite
Given two positive functions $fcolon mathbb{R} rightarrow mathbb{R}^{+}$ and $gcolon mathbb{R} rightarrow mathbb{R}^{+}$ and a point $ x in mathbb{R}$.
Is there some sufficient (and preferably necessary) criterion asserting that every $n$-th derivative of $g$ dominates the respective $n$-th derivative of $f$ at point $x$?
I.e. the criterion should assert that
$$
forall ncolon quad f^{(n)}(x) leq g^{(n)}(x)~.
$$
(Needless to say, the criterion should be somehow easier to check than the above.)
real-analysis derivatives
asked Nov 21 at 23:36
blk
356
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Given two positive functions $fcolon mathbb{R} rightarrow mathbb{R}^{+}$ and $gcolon mathbb{R} rightarrow mathbb{R}^{+}$ and a point $ x in mathbb{R}$.
Is there some sufficient (and preferably necessary) criterion asserting that every $n$-th derivative of $g$ dominates the respective $n$-th derivative of $f$ at point $x$?
I.e. the criterion should assert that
$$
forall ncolon quad f^{(n)}(x) leq g^{(n)}(x)~.
$$
(Needless to say, the criterion should be somehow easier to check than the above.)
real-analysis derivatives
asked Nov 21 at 23:36
blk
356
Given two positive functions $fcolon mathbb{R} rightarrow mathbb{R}^{+}$ and $gcolon mathbb{R} rightarrow mathbb{R}^{+}$ and a point $ x in mathbb{R}$.
Is there some sufficient (and preferably necessary) criterion asserting that every $n$-th derivative of $g$ dominates the respective $n$-th derivative of $f$ at point $x$?
I.e. the criterion should assert that
$$
forall ncolon quad f^{(n)}(x) leq g^{(n)}(x)~.
$$
(Needless to say, the criterion should be somehow easier to check than the above.)
real-analysis derivatives
real-analysis derivatives
asked Nov 21 at 23:36
blk
356
asked Nov 21 at 23:36
blk
356
asked Nov 21 at 23:36
blk
356
asked Nov 21 at 23:36
blk
356
asked Nov 21 at 23:36
blk
356
356
add a comment |
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008537%2fcriterion-for-all-derivatives-of-a-function-dominating-the-derivatives-of-anothe%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3008537%2fcriterion-for-all-derivatives-of-a-function-dominating-the-derivatives-of-anothe%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
