function postsbycategory() {
// the query
$the_query = new WP_Query( array( 'category_name' => 'karan-johar', 'posts_per_page' => 6 ) ); 

$i=1;
// The Loop
if ( $the_query->have_posts() ) {
$string .= '<ul class="postsbycategory widget_recent_entries">';
while ( $the_query->have_posts() ) {
    $the_query->the_post();
    if($i>1){
        if ( has_post_thumbnail() ) {
        $string .= '<li>';
        $string .= '<a href="' . get_the_permalink() .'" rel="bookmark">' . get_the_post_thumbnail($post_id, array( 50, 50) ) . get_the_title() .'</a></li>';
        } else { 
        // if no featured image is found
        $string .= '<li><a href="' . get_the_permalink() .'" rel="bookmark">' . get_the_title() .'</a></li>';
        }
        }
        ++$i;
        }
} else {
// no posts found
}
$string .= '</ul>';

return $string;
}