Creating a basic secure php mailer

I am busy creating a basic php mailer script to post to _self and email to a address.

Is the script secure?

How can I avoid someone clicking on submit the whole time, to spam the mailbox, with minimal extra code

<?php
//Mail header removal
function remove_headers($string) { 
   $headers = array(
   "/to:/i",
   "/from:/i",
   "/bcc:/i",
   "/cc:/i",
   "/Content-Transfer-Encoding:/i",
   "/Content-Type:/i",
   "/Mime-Version:/i" 
 ); 
$string = preg_replace($headers, '', $string);
return strip_tags($string);
} 

$to      = "email@tosendto.com";
$subject = "Sent from site";

$uname    = remove_headers($_POST['fname']);
$uemail   = remove_headers($_POST['femail']);
$umessage = remove_headers($_POST['fmessage']);
$umessage = "Name : " . $uname . " Email : " . $uemail . " Message : " . $umessage;

if(isset($_POST['submit']))
{
   mail($to, $subject, $umessage, "From: page@website.com");
}   

?>

<div id="mailer" >
  <h1>Message</h1>
  <form name="test" action="<?php echo htmlentities($_SERVER['PHP_SELF']); ?>" method="post">
<p>Your Name:</p>
<input type="text" size="20" name="fname"><br><br>
<p>Your Email:</p>
<input type="text" size="20" name="femail"><br><br>
<p>Your Message:</p>
<textarea name="fmessage" rows="4" cols="20"></textarea><br><br>
<input type="submit" name="submit" value="Send Message">
  </form>
  <?php if(isset($_POST['submit']))
  {
   echo "<p>Sent. We will be in contact shortly.</p>";
  } ?>

  </div>

Your using the email sanitize filter on each field, you need to use FILTER_SANITIZE_STRING for the name and FILTER_SANITIZE_FULL_SPECIAL_CHARS for the message field.

Sanitizing is not the same as validating…

VALIDATE Filters

SANITIZE filters