secure php contact form with htmlspecialchars($_SERVER["PHP_SELF"]) - php

Question

I am very new to php and am trying to create a simple, secure and functioning contact form on a webpage. My form is as followed:
<form class="form-horizontal" method="post" action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]);?>" >
<fieldset>
<div class="form-group">
<span class="col-md-1 text-center"><i class="fa fa-user bigicon"></i></span>
<div class="col-md-11">
<input id="fname" name="name" type="text" placeholder="Name" class="form-control">
</div>
</div>
<div class="form-group">
<span class="col-md-1 text-center"><i class="fa fa-envelope-o bigicon"></i></span>
<div class="col-md-11">
<input id="email" name="email" type="text" placeholder="Email Address" class="form-control">
</div>
</div>
<div class="form-group">
<span class="col-md-1 text-center"><i class="fa fa-pencil-square-o bigicon"></i></span>
<div class="col-md-11">
<textarea class="form-control" id="message" name="message" placeholder="Enter your massage for us here. We will get back to you within 2 business days." rows="7"></textarea>
</div>
</div>
<div class="form-group">
<div class="col-md-12 text-right">
<button type="submit" class="btn btn-primary btn-lg">Send</button>
</div>
</div>
</fieldset>
and I have the following php file (called mail.php) that I would like to execute when SEND button is pressed.
<?php
$name = $_POST['name'];
$email = $_POST['email'];
$message = $_POST['message'];
$formcontent="From: $name \n Message: $message";
$recipient = "some.email#gmail.com";
$subject = "Contact Form";
$mailheader = "From: $email \r\n";
mail($recipient, $subject, $formcontent, $mailheader) or die("Error!");?>
Question is; if I have
action="<?php echo htmlspecialchars($_SERVER["PHP_SELF"]);?>"
how can I execute the php script in mail.php?