Hestia theme

Hestia theme doesn't really have a child theme friendly header, as it seems like their header replacement is a Pro version feature, so currently you can only replace their code by modifying the original PHP file (what you would need to modify again after updating). If that is fine for you, open up this file:

/wp-content/themes/hestia/inc/views/front-page/class-hestia-big-title-section.php

It will have this code:

public function render_content() {
    $section_content      = $this->get_big_title_content();
    $big_title_background = $this->get_big_title_background();

    if ( empty( $big_title_background ) && empty( $section_content ) ) {
        return;
    } ?>

    <div class="item active">
        <div class="page-header">
            <?php
            if ( is_customize_preview() ) {
                echo '<div class="big-title-image"></div>';
            }
            hestia_before_big_title_section_content_trigger();
            ?>

            <div class="container">
                <?php hestia_top_big_title_section_content_trigger(); ?>
                <div class="row hestia-big-title-content">
                    <?php $this->show_big_title_content( $section_content ); ?>
                </div>
                <?php hestia_bottom_big_title_section_content_trigger(); ?>
            </div><!-- /.container -->

            <div class="header-filter"
                <?php
                if ( ! empty( $big_title_background ) ) {
                    echo 'style="background-image: url(' . esc_url( $big_title_background ) . ')"';
                }
                ?>
            ></div><!-- /.header-filter -->
            <?php hestia_after_big_title_section_content_trigger(); ?>
        </div><!-- /.page-header -->
    </div>
    <?php
}

Replace it to use our slider's shortcode instead:

public function render_content() {
	echo do_shortcode('[smartslider3 slider=316]');
}

If you don't want to modify the theme's codes like this, then contact the Hestia developers to ask them wether or not the Pro version still offers the feature of replacing their header with a shortcode, and if it does, consider purchasing their theme.